Poço Quadrado Finito

Neste documento demonstramos a utilização do software e indicamos os resultados, que podem ser visualizados no modo abaixo ou acessados na pasta saidas.

Bibliotecas utilizadas

In [1]:
# python standard
import time

# third-party
import pandas as pd
import numpy as np

# locals
from core.constantes import *
from core.poco_quadrado_finito import PocoQuantico

Configurações gráficas

In [2]:
import locale
locale.setlocale(locale.LC_NUMERIC, "pt_BR.UTF-8")

import matplotlib.style
import matplotlib as mpl
mpl.style.use('classic')

%matplotlib inline
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator
plt.style.use('mestrado')

Inicia um poço quântico

In [3]:
poco_100 = PocoQuantico(well_length=100.0, well_height=1.0, N=2048, dt=1e-18)

Este é o potencial que vamos usar:

In [4]:
fix, ax = plt.subplots(1,1, figsize=(18,6))
ax.plot(poco_100.z_ang, poco_100.v_ev)
ax.set_xlabel(r'z (A)')
ax.set_ylabel(r'V (eV)')
plt.show()

que foi iniciado com as seguintes propriedades:

  • well_length=100.0, largura de 100.0 Angstrom
  • well_height=1.0, altura de 1.0 eV
  • N=2048, 2048 pontos na malha (discretização)
  • dt=1e-19, passo de tempo de $1 \times 10^{-18}$ segundos

Calculo dos Autoestados e Autovalores

O comando abaxo solicita o cálculo dos autoestados e autovalores com precisao de 0.01 %.

Está precisão não é em relação ao valor analítico, mas em relação ao valor para o qual o algoritmo convergiu (que possui um erro em relação ao analítico).

In [6]:
poco_100.evolucao_imaginaria(precision=1e-4)
Energy [0]:
                                Numeric=3.4898948254e-03
                                Analytic=3.4828963796e-03
                                iterations=104000
Energy [1]:
                                Numeric=1.3951814274e-02
                                Analytic=1.3929748482e-02
                                iterations=76000
Energy [2]:
                                Numeric=3.1373388323e-02
                                Analytic=3.1334961728e-02
                                iterations=48000
Energy [3]:
                                Numeric=5.5750644769e-02
                                Analytic=5.5688922460e-02
                                iterations=41000
Energy [4]:
                                Numeric=8.7064283660e-02
                                Analytic=8.6977528175e-02
                                iterations=31000
Energy [5]:
                                Numeric=1.2528953568e-01
                                Analytic=1.2518144938e-01
                                iterations=29000
Energy [6]:
                                Numeric=1.7042458894e-01
                                Analytic=1.7027502006e-01
                                iterations=23000
Energy [7]:
                                Numeric=2.2240967245e-01
                                Analytic=2.2222458156e-01
                                iterations=22000
Energy [8]:
                                Numeric=2.8119865520e-01
                                Analytic=2.8098597641e-01
                                iterations=19000
Energy [9]:
                                Numeric=3.4675936894e-01
                                Analytic=3.4650064353e-01
                                iterations=18000
Energy [10]:
                                Numeric=4.1898720865e-01
                                Analytic=4.1868926581e-01
                                iterations=16000
Energy [11]:
                                Numeric=4.9778467623e-01
                                Analytic=4.9744080362e-01
                                iterations=15000
Energy [12]:
                                Numeric=5.8299325437e-01
                                Analytic=5.8259197003e-01
                                iterations=13000
Energy [13]:
                                Numeric=6.7429257248e-01
                                Analytic=6.7388421325e-01
                                iterations=13000
Energy [14]:
                                Numeric=7.7127282048e-01
                                Analytic=7.7085693460e-01
                                iterations=12000
Energy [15]:
                                Numeric=8.7295472920e-01
                                Analytic=8.7249408531e-01
                                iterations=12000
Energy [16]:
                                Numeric=9.7561504625e-01
                                Analytic=9.7487073823e-01
                                iterations=15000
Out[6]:
<__main__.PocoQuantico at 0x7fb7e1a47c18>

Como é possível ver acima (clique na tela para rolar) temos 17 autoestados, com autovalores:

In [7]:
print(poco_100.eigenvalues)
[ 0.00348989  0.01395181  0.03137339  0.05575064  0.08706428  0.12528954
  0.17042459  0.22240967  0.28119866  0.34675937  0.41898721  0.49778468
  0.58299325  0.67429257  0.77127282  0.87295473  0.97561505]

Os autovalores analíticos são:

In [8]:
poco_100.analytical_eigenvalues = np.array(poco_100.solucao_analitica())
print(poco_100.analytical_eigenvalues)
[ 0.0034829   0.01392975  0.03133496  0.05568892  0.08697753  0.12518145
  0.17027502  0.22222458  0.28098598  0.34650064  0.41868927  0.4974408
  0.58259197  0.67388421  0.77085693  0.87249409  0.97487074]

Vamos ver o erro em cada autovalor:

In [9]:
eigenvalues = pd.DataFrame({'analytic': poco_100.analytical_eigenvalues, 'numeric': poco_100.eigenvalues})
eigenvalues['error (%)'] = eigenvalues.apply(lambda l: 100.0*(l['numeric']/l['analytic']-1.0), axis=1)
print(eigenvalues)
    analytic   numeric  error (%)
0   0.003483  0.003490   0.200938
1   0.013930  0.013952   0.158408
2   0.031335  0.031373   0.122632
3   0.055689  0.055751   0.110834
4   0.086978  0.087064   0.099745
5   0.125181  0.125290   0.086344
6   0.170275  0.170425   0.087840
7   0.222225  0.222410   0.083290
8   0.280986  0.281199   0.075690
9   0.346501  0.346759   0.074668
10  0.418689  0.418987   0.071161
11  0.497441  0.497785   0.069128
12  0.582592  0.582993   0.068879
13  0.673884  0.674293   0.060598
14  0.770857  0.771273   0.053951
15  0.872494  0.872955   0.052796
16  0.974871  0.975615   0.076349

Vamos imprimir os autoestados:

In [10]:
fix, axs = plt.subplots(5,4, figsize=(22,22))
for i in range(5):
    for j in range(4):
        if (4*i+j) >= len(poco_100.eigenstates):
            continue
        axs[i,j].plot(poco_100.z_ang, np.abs(poco_100.eigenstates[4*i+j])**2)
        axs[i,j].set_xlabel(r'z (A)')
        axs[i,j].set_ylabel(r'$|\psi_{%d} (z)|^2$' % (4*i+j))
plt.show()

Estes autoestados estão normalizados para um eixo z em unidades atômicas, vamos normalizar eles para um eixo z em Angstrom:

In [11]:
z = poco_100.z_ang
def normalize2ang(psi):
    # psi / (<psi|psi>)^(1/2)
    return psi / np.sqrt(simps(psi.conj()*psi,z))
poco_100.eigenstates_ang = [normalize2ang(es) for es in poco_100.eigenstates]

vamos imprimir para ver que está tudo certo:

In [12]:
fix, axs = plt.subplots(5,4, figsize=(22,22))
for i in range(5):
    for j in range(4):
        if (4*i+j) >= len(poco_100.eigenstates_ang):
            continue
        axs[i,j].plot(poco_100.z_ang, np.abs(poco_100.eigenstates_ang[4*i+j])**2)
        axs[i,j].set_xlabel(r'z (A)')
        axs[i,j].set_ylabel(r'$|\psi_{%d} (z)|^2$' % (4*i+j))
plt.show()

Vamos colocar os dados em um DataFrame para depois exportá-los para um .csv.

In [13]:
export_values = {
    'z_ang': poco_100.z_ang, # o eixo z em Angstrom
    'v_ev': poco_100.v_ev, # o potencial em eV
}

for i, state in enumerate(poco_100.eigenstates_ang):
    export_values['psi_%02d' % i] = state # psi
    export_values['psi_squared_%d' % i] = np.abs(state)**2 # |psi|^2
results = pd.DataFrame(export_values)
results.to_csv('saidas/poco_quantico_estados.csv')
results.head()
Out[13]:
psi_00 psi_01 psi_02 psi_03 psi_04 psi_05 psi_06 psi_07 psi_08 psi_09 ... psi_squared_2 psi_squared_3 psi_squared_4 psi_squared_5 psi_squared_6 psi_squared_7 psi_squared_8 psi_squared_9 v_ev z_ang
0 (-4.82294817881e-15+1.12964096718e-15j) (-3.29406817957e-12-9.23908767427e-17j) (1.89685488369e-15-1.5486171481e-15j) (-6.65684309279e-12+1.62156548197e-17j) (2.58336684144e-15+1.71939908397e-15j) (-1.00738141044e-11-1.08157349786e-15j) (9.0381653665e-15-1.55288386129e-15j) (-1.34261504582e-11+4.84283761882e-16j) (1.25868856246e-14+7.83583350216e-16j) (-1.67870529003e-11+2.10897374484e-16j) ... 5.996274e-30 4.431356e-23 9.630117e-30 1.014817e-22 8.409988e-29 1.802615e-22 1.590437e-28 2.818051e-22 1.0 -150.000000
1 (-1.05227960265e-14+1.03796143012e-15j) (3.29422902649e-12-9.55842216151e-17j) (-1.57371978487e-14-1.47538535868e-15j) (6.65654736869e-12-3.23319078479e-17j) (-2.68441416418e-14+1.59549638385e-15j) (1.00753716049e-11-1.05032590027e-15j) (-3.25773971547e-14-1.47869246396e-15j) (1.34272094008e-11+4.88771433964e-16j) (-4.09073678505e-14+7.12970155475e-16j) (1.67897970222e-11+3.05038815012e-16j) ... 2.498362e-28 4.430962e-23 7.231535e-28 1.015131e-22 1.063473e-27 1.802900e-22 1.673921e-27 2.818973e-22 1.0 -149.853444
2 (1.22003432191e-15+9.5089444218e-16j) (-3.29416290053e-12-9.01317914318e-17j) (1.94546520686e-14-1.43497471378e-15j) (-6.65692299997e-12-7.55440994737e-17j) (3.23635863828e-14+1.50051379996e-15j) (-1.00741501524e-11-9.7983700616e-16j) (5.03772429289e-14-1.36916856347e-15j) (-1.34267076408e-11+4.52422542553e-16j) (6.63290640326e-14+6.35701013715e-16j) (-1.67875923063e-11+3.61441494123e-16j) ... 3.805426e-28 4.431462e-23 1.049653e-27 1.014885e-22 2.539741e-27 1.802765e-22 4.399949e-27 2.818233e-22 1.0 -149.706888
3 (-1.60701395839e-14+8.92398761367e-16j) (3.29439463925e-12-1.14971940227e-16j) (-3.33998464234e-14-1.35396626355e-15j) (6.65693848969e-12-9.10137357047e-17j) (-5.618585553e-14+1.38946754031e-15j) (1.00758083477e-11-9.39206819295e-16j) (-7.4288299234e-14-1.32647489479e-15j) (1.34277016281e-11+4.30129757794e-16j) (-9.43444077801e-14+6.02188072166e-16j) (1.67905319264e-11+3.77626964146e-16j) ... 1.117383e-27 4.431483e-23 3.158781e-27 1.015219e-22 5.520511e-27 1.803032e-22 8.901230e-27 2.819220e-22 1.0 -149.560332
4 (7.23442226821e-15+8.23075411143e-16j) (-3.29435591684e-12-9.90009876382e-17j) (3.70124492678e-14-1.31290057687e-15j) (-6.65729237544e-12-8.88866409581e-17j) (6.21056750103e-14+1.30348842182e-15j) (-1.00748359522e-11-8.57115115791e-16j) (9.17544563236e-14-1.21613583764e-15j) (-1.34277087815e-11+3.82003362191e-16j) (1.20033100157e-13+5.38664544098e-16j) (-1.67887176089e-11+3.68970096635e-16j) ... 1.371645e-27 4.431954e-23 3.858814e-27 1.015023e-22 8.420359e-27 1.803034e-22 1.440824e-26 2.818610e-22 1.0 -149.413776

5 rows × 36 columns

Nós temos também informações sobre tempo de processamento e número de iterações para cada nível de energia:

In [14]:
perf_metrics = pd.DataFrame({
    'time (s)': poco_100.timers,
    'iterations': poco_100.counters,
    'numeric precision (%)': 100*poco_100.eigenvalues_precisions,
    'analytic (eV)': poco_100.analytical_eigenvalues,
    'numeric (eV)': poco_100.eigenvalues,
    'actual precision (%)': 100.0*(poco_100.eigenvalues/poco_100.analytical_eigenvalues-1.0)
})
perf_metrics.to_csv('saidas/poco_quantico_metricas.csv')
perf_metrics
Out[14]:
actual precision (%) analytic (eV) iterations numeric (eV) numeric precision (%) time (s)
0 0.200938 0.003483 104000 0.003490 0.009897 79.097671
1 0.158408 0.013930 76000 0.013952 0.009482 117.730071
2 0.122632 0.031335 48000 0.031373 0.008770 108.682453
3 0.110834 0.055689 41000 0.055751 0.009870 115.231905
4 0.099745 0.086978 31000 0.087064 0.009191 108.636536
5 0.086344 0.125181 29000 0.125290 0.007457 120.827942
6 0.087840 0.170275 23000 0.170425 0.009140 112.861944
7 0.083290 0.222225 22000 0.222410 0.007595 119.852372
8 0.075690 0.280986 19000 0.281199 0.006839 120.395632
9 0.074668 0.346501 18000 0.346759 0.006870 121.071307
10 0.071161 0.418689 16000 0.418987 0.006269 96.079016
11 0.069128 0.497441 15000 0.497785 0.007496 87.088524
12 0.068879 0.582592 13000 0.582993 0.009941 114.925910
13 0.060598 0.673884 13000 0.674293 0.007896 123.765117
14 0.053951 0.770857 12000 0.771273 0.008039 122.778017
15 0.052796 0.872494 12000 0.872955 0.008532 118.763190
16 0.076349 0.974871 15000 0.975615 0.008492 160.496706

Augumas relações interessantes

In [15]:
metrics = perf_metrics.copy()
metrics.reset_index(inplace=True)
ax = metrics[['iterations', 'time (s)']].plot(secondary_y='time (s)', figsize=(18,10))
ax.set_xlabel('eigenstate number')
ax.set_ylabel('Numero de iteracoes')
ax.right_ax.set_ylabel('Tempo de processamento (s)')
plt.show()
In [16]:
ax = metrics[['numeric precision (%)', 'actual precision (%)']].plot(secondary_y='actual precision (%)', figsize=(18,10))
ax.set_xlabel('eigenstate number')
ax.set_ylabel('Precisao Numerica (%)')
ax.right_ax.set_ylabel('Erro em relacao ao Valor Analitico (%)')
plt.show()

Verificando os resultados analíticos

Aqui vamos variar o dispositivo em largura e verificar os autovalores obtidos analiticamente.

In [17]:
eigenvalues = []
eigenvalues_per_length = []
for Vb in [1.0]:
    for L in np.linspace(10, 80, 100):
        d = PocoQuantico(well_length=L, well_height=Vb)
        evs = d.solucao_analitica()
        for ev in evs:
            eigenvalues.append((L,ev))
        eigenvalues_per_length.append((L, len(evs)))
        print("finalizamos %.2f" % L)
finalizamos 10.00
finalizamos 10.71
finalizamos 11.41
finalizamos 12.12
finalizamos 12.83
finalizamos 13.54
finalizamos 14.24
finalizamos 14.95
finalizamos 15.66
finalizamos 16.36
finalizamos 17.07
finalizamos 17.78
finalizamos 18.48
finalizamos 19.19
finalizamos 19.90
finalizamos 20.61
finalizamos 21.31
finalizamos 22.02
finalizamos 22.73
finalizamos 23.43
finalizamos 24.14
finalizamos 24.85
finalizamos 25.56
finalizamos 26.26
finalizamos 26.97
finalizamos 27.68
finalizamos 28.38
finalizamos 29.09
finalizamos 29.80
finalizamos 30.51
finalizamos 31.21
finalizamos 31.92
finalizamos 32.63
finalizamos 33.33
finalizamos 34.04
finalizamos 34.75
finalizamos 35.45
finalizamos 36.16
finalizamos 36.87
finalizamos 37.58
finalizamos 38.28
finalizamos 38.99
finalizamos 39.70
finalizamos 40.40
finalizamos 41.11
finalizamos 41.82
finalizamos 42.53
finalizamos 43.23
finalizamos 43.94
finalizamos 44.65
finalizamos 45.35
finalizamos 46.06
finalizamos 46.77
finalizamos 47.47
finalizamos 48.18
finalizamos 48.89
finalizamos 49.60
finalizamos 50.30
finalizamos 51.01
finalizamos 51.72
finalizamos 52.42
finalizamos 53.13
finalizamos 53.84
finalizamos 54.55
finalizamos 55.25
finalizamos 55.96
finalizamos 56.67
finalizamos 57.37
finalizamos 58.08
finalizamos 58.79
finalizamos 59.49
finalizamos 60.20
finalizamos 60.91
finalizamos 61.62
finalizamos 62.32
finalizamos 63.03
finalizamos 63.74
finalizamos 64.44
finalizamos 65.15
finalizamos 65.86
finalizamos 66.57
finalizamos 67.27
finalizamos 67.98
finalizamos 68.69
finalizamos 69.39
finalizamos 70.10
finalizamos 70.81
finalizamos 71.52
finalizamos 72.22
finalizamos 72.93
finalizamos 73.64
finalizamos 74.34
finalizamos 75.05
finalizamos 75.76
finalizamos 76.46
finalizamos 77.17
finalizamos 77.88
finalizamos 78.59
finalizamos 79.29
finalizamos 80.00
In [18]:
fix, ax = plt.subplots(1,1, figsize=(18,14))
x,y=tuple(zip(*eigenvalues))
ax.scatter(x,y, marker='.')
plt.show()
In [19]:
fix, ax = plt.subplots(1,1, figsize=(18,10))
x,y=tuple(zip(*eigenvalues_per_length))
ax.scatter(x,y)
plt.show()

vamos exportar os valores

In [20]:
L,e=tuple(zip(*eigenvalues))
l_e = pd.DataFrame({'L':L, 'e':e})
l_e.to_csv('saidas/poco_quantico_autovalores_por_comprimento.csv')
#l_e
In [21]:
L,n=tuple(zip(*eigenvalues_per_length))
l_n = pd.DataFrame({'L':L, 'n':n})
l_n.to_csv('saidas/poco_quantico_numero_de_autovalores_por_comprimento.csv')
#l_n
In [22]:
eigenvalues = []
eigenvalues_per_length = []
for Vb in [1.0]:
    for L in np.linspace(10, 80, 100):
        d = PocoQuantico(well_length=L, well_height=Vb, N=2048, dt=1e-18)
        d.evolucao_imaginaria(precision=1e-4)
        evs = d.eigenvalues
        for ev in evs:
            eigenvalues.append((L,ev))
        eigenvalues_per_length.append((L, len(evs)))
        print('Estamos em L = %.2f' % L)
        
L,e=tuple(zip(*eigenvalues))
l_e = pd.DataFrame({'L':L, 'e':e})
l_e.to_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv')
Energy [0]:
                                Numeric=1.9087770884e-01
                                Analytic=1.9076054414e-01
                                iterations=5000
Energy [1]:
                                Numeric=7.0451637873e-01
                                Analytic=7.0276957111e-01
                                iterations=5000
Estamos em L = 10.00
Energy [0]:
                                Numeric=1.7334442584e-01
                                Analytic=1.7322678903e-01
                                iterations=5000
Energy [1]:
                                Numeric=6.4824094252e-01
                                Analytic=6.4731212760e-01
                                iterations=5000
Estamos em L = 10.71
Energy [0]:
                                Numeric=1.5807630540e-01
                                Analytic=1.5796347786e-01
                                iterations=5000
Energy [1]:
                                Numeric=5.9741061257e-01
                                Analytic=5.9681360721e-01
                                iterations=5000
Estamos em L = 11.41
Energy [0]:
                                Numeric=1.4470971004e-01
                                Analytic=1.4460324525e-01
                                iterations=5000
Energy [1]:
                                Numeric=5.5155953861e-01
                                Analytic=5.5110359698e-01
                                iterations=5000
Estamos em L = 12.12
Energy [0]:
                                Numeric=1.3294818095e-01
                                Analytic=1.3284809547e-01
                                iterations=5000
Energy [1]:
                                Numeric=5.1022850498e-01
                                Analytic=5.0983916870e-01
                                iterations=5000
Energy [2]:
                                Numeric=9.7844069033e-01
                                Analytic=9.8552408029e-01
                                iterations=16000
Estamos em L = 12.83
Energy [0]:
                                Numeric=1.2254765693e-01
                                Analytic=1.2245482543e-01
                                iterations=6000
Energy [1]:
                                Numeric=4.7296750814e-01
                                Analytic=4.7261538262e-01
                                iterations=5000
Energy [2]:
                                Numeric=9.4311265450e-01
                                Analytic=9.4695789904e-01
                                iterations=14000
Estamos em L = 13.54
Energy [0]:
                                Numeric=1.1331069022e-01
                                Analytic=1.1322372030e-01
                                iterations=6000
Energy [1]:
                                Numeric=4.3934810356e-01
                                Analytic=4.3902109138e-01
                                iterations=5000
Energy [2]:
                                Numeric=8.9951495650e-01
                                Analytic=9.0062555928e-01
                                iterations=11000
Estamos em L = 14.24
Energy [0]:
                                Numeric=1.0507156265e-01
                                Analytic=1.0498977229e-01
                                iterations=6000
Energy [1]:
                                Numeric=4.0896879556e-01
                                Analytic=4.0866592077e-01
                                iterations=6000
Energy [2]:
                                Numeric=8.5249581616e-01
                                Analytic=8.5250675297e-01
                                iterations=9000
Estamos em L = 14.95
Energy [0]:
                                Numeric=9.7692079919e-02
                                Analytic=9.7615835088e-02
                                iterations=7000
Energy [1]:
                                Numeric=3.8147770720e-01
                                Analytic=3.8119200267e-01
                                iterations=6000
Energy [2]:
                                Numeric=8.0552166579e-01
                                Analytic=8.0517558023e-01
                                iterations=8000
Estamos em L = 15.66
Energy [0]:
                                Numeric=9.1059086969e-02
                                Analytic=9.0987262182e-02
                                iterations=7000
Energy [1]:
                                Numeric=3.5654792794e-01
                                Analytic=3.5627770760e-01
                                iterations=6000
Energy [2]:
                                Numeric=7.6027571508e-01
                                Analytic=7.5981577825e-01
                                iterations=7000
Estamos em L = 16.36
Energy [0]:
                                Numeric=8.5075688988e-02
                                Analytic=8.5007685233e-02
                                iterations=7000
Energy [1]:
                                Numeric=3.3389335704e-01
                                Analytic=3.3363729016e-01
                                iterations=6000
Energy [2]:
                                Numeric=7.1744415556e-01
                                Analytic=7.1695055514e-01
                                iterations=6000
Estamos em L = 17.07
Energy [0]:
                                Numeric=7.9660580891e-02
                                Analytic=7.9595671364e-02
                                iterations=7000
Energy [1]:
                                Numeric=3.1326171735e-01
                                Analytic=3.1301857022e-01
                                iterations=6000
Energy [2]:
                                Numeric=6.7724417228e-01
                                Analytic=6.7676474498e-01
                                iterations=6000
Estamos em L = 17.78
Energy [0]:
                                Numeric=7.4742420431e-02
                                Analytic=7.4682061157e-02
                                iterations=8000
Energy [1]:
                                Numeric=2.9443130471e-01
                                Analytic=2.9419979640e-01
                                iterations=6000
Energy [2]:
                                Numeric=6.3972772356e-01
                                Analytic=6.3926404915e-01
                                iterations=6000
Energy [3]:
                                Numeric=1.0146771640e+00
                                Analytic=9.9953527480e-01
                                iterations=14000
Estamos em L = 18.48
Energy [0]:
                                Numeric=7.0265462997e-02
                                Analytic=7.0207836528e-02
                                iterations=8000
Energy [1]:
                                Numeric=2.7720428753e-01
                                Analytic=2.7698630086e-01
                                iterations=7000
Energy [2]:
                                Numeric=6.0480494101e-01
                                Analytic=6.0435942918e-01
                                iterations=6000
Energy [3]:
                                Numeric=9.8172408479e-01
                                Analytic=9.7791008155e-01
                                iterations=13000
Estamos em L = 19.19
Energy [0]:
                                Numeric=6.6176393151e-02
                                Analytic=6.6122403312e-02
                                iterations=9000
Energy [1]:
                                Numeric=2.6141472421e-01
                                Analytic=2.6120725666e-01
                                iterations=7000
Energy [2]:
                                Numeric=5.7234233252e-01
                                Analytic=5.7191381150e-01
                                iterations=6000
Energy [3]:
                                Numeric=9.4516355070e-01
                                Analytic=9.4374687345e-01
                                iterations=11000
Estamos em L = 19.90
Energy [0]:
                                Numeric=6.2433831333e-02
                                Analytic=6.2382200205e-02
                                iterations=9000
Energy [1]:
                                Numeric=2.4691062584e-01
                                Analytic=2.4671268467e-01
                                iterations=7000
Energy [2]:
                                Numeric=5.4217948172e-01
                                Analytic=5.4176858088e-01
                                iterations=6000
Energy [3]:
                                Numeric=9.0677023073e-01
                                Analytic=9.0595404030e-01
                                iterations=10000
Estamos em L = 20.61
Energy [0]:
                                Numeric=5.8999372120e-02
                                Analytic=5.8949565955e-02
                                iterations=9000
Energy [1]:
                                Numeric=2.3356030754e-01
                                Analytic=2.3337076703e-01
                                iterations=7000
Energy [2]:
                                Numeric=5.1415230715e-01
                                Analytic=5.1375874376e-01
                                iterations=6000
Energy [3]:
                                Numeric=8.6814979351e-01
                                Analytic=8.6749062249e-01
                                iterations=9000
Estamos em L = 21.31
Energy [0]:
                                Numeric=5.5838399930e-02
                                Analytic=5.5791812061e-02
                                iterations=10000
Energy [1]:
                                Numeric=2.2124440237e-01
                                Analytic=2.2106547621e-01
                                iterations=8000
Energy [2]:
                                Numeric=4.8810007366e-01
                                Analytic=4.8772154005e-01
                                iterations=6000
Energy [3]:
                                Numeric=8.3027603352e-01
                                Analytic=8.2965954663e-01
                                iterations=8000
Estamos em L = 22.02
Energy [0]:
                                Numeric=5.2925470487e-02
                                Analytic=5.2880459850e-02
                                iterations=10000
Energy [1]:
                                Numeric=2.0986572187e-01
                                Analytic=2.0969450457e-01
                                iterations=8000
Energy [2]:
                                Numeric=4.6386405273e-01
                                Analytic=4.6350118696e-01
                                iterations=6000
Energy [3]:
                                Numeric=7.9369415684e-01
                                Analytic=7.9308939304e-01
                                iterations=7000
Estamos em L = 22.73
Energy [0]:
                                Numeric=5.0232902498e-02
                                Analytic=5.0190609763e-02
                                iterations=11000
Energy [1]:
                                Numeric=1.9933191304e-01
                                Analytic=1.9916746783e-01
                                iterations=8000
Energy [2]:
                                Numeric=4.4130071791e-01
                                Analytic=4.4095131236e-01
                                iterations=6000
Energy [3]:
                                Numeric=7.5865051612e-01
                                Analytic=7.5808247621e-01
                                iterations=7000
Estamos em L = 23.43
Energy [0]:
                                Numeric=4.7741373832e-02
                                Analytic=4.7700417507e-02
                                iterations=11000
Energy [1]:
                                Numeric=1.8956316286e-01
                                Analytic=1.8940435241e-01
                                iterations=8000
Energy [2]:
                                Numeric=4.2027173473e-01
                                Analytic=4.1993600366e-01
                                iterations=6000
Energy [3]:
                                Numeric=7.2530969843e-01
                                Analytic=7.2476781838e-01
                                iterations=7000
Estamos em L = 24.14
Energy [0]:
                                Numeric=4.5430763809e-02
                                Analytic=4.5390657015e-02
                                iterations=11000
Energy [1]:
                                Numeric=1.8048374270e-01
                                Analytic=1.8033417623e-01
                                iterations=9000
Energy [2]:
                                Numeric=4.0065288109e-01
                                Analytic=4.0033003326e-01
                                iterations=6000
Energy [3]:
                                Numeric=6.9370188194e-01
                                Analytic=6.9317757857e-01
                                iterations=7000
Energy [4]:
                                Numeric=9.9406640139e-01
                                Analytic=9.9545379356e-01
                                iterations=33000
Estamos em L = 24.85
Energy [0]:
                                Numeric=4.3281847119e-02
                                Analytic=4.3244354258e-02
                                iterations=12000
Energy [1]:
                                Numeric=1.7203803344e-01
                                Analytic=1.7189383463e-01
                                iterations=9000
Energy [2]:
                                Numeric=3.8232528318e-01
                                Analytic=3.8201860448e-01
                                iterations=7000
Energy [3]:
                                Numeric=6.6379388157e-01
                                Analytic=6.6328855024e-01
                                iterations=7000
Energy [4]:
                                Numeric=9.7190460266e-01
                                Analytic=9.7158004037e-01
                                iterations=19000
Estamos em L = 25.56
Energy [0]:
                                Numeric=4.1283290383e-02
                                Analytic=4.1246479170e-02
                                iterations=12000
Energy [1]:
                                Numeric=1.6416694982e-01
                                Analytic=1.6402710597e-01
                                iterations=9000
Energy [2]:
                                Numeric=3.6519110117e-01
                                Analytic=3.6489683037e-01
                                iterations=7000
Energy [3]:
                                Numeric=6.3553282516e-01
                                Analytic=6.3504598865e-01
                                iterations=7000
Energy [4]:
                                Numeric=9.4214178018e-01
                                Analytic=9.4146885537e-01
                                iterations=13000
Estamos em L = 26.26
Energy [0]:
                                Numeric=3.9418200905e-02
                                Analytic=3.9383685421e-02
                                iterations=13000
Energy [1]:
                                Numeric=1.5681568181e-01
                                Analytic=1.5668379406e-01
                                iterations=10000
Energy [2]:
                                Numeric=3.4915353236e-01
                                Analytic=3.4886907686e-01
                                iterations=7000
Energy [3]:
                                Numeric=6.0885088613e-01
                                Analytic=6.0837781652e-01
                                iterations=7000
Energy [4]:
                                Numeric=9.1025777369e-01
                                Analytic=9.0954836051e-01
                                iterations=10000
Estamos em L = 26.97
Energy [0]:
                                Numeric=3.7678091628e-02
                                Analytic=3.7644089781e-02
                                iterations=13000
Energy [1]:
                                Numeric=1.4994671853e-01
                                Analytic=1.4981898803e-01
                                iterations=10000
Energy [2]:
                                Numeric=3.3412217372e-01
                                Analytic=3.3384825084e-01
                                iterations=7000
Energy [3]:
                                Numeric=5.8365984090e-01
                                Analytic=5.8320329872e-01
                                iterations=7000
Energy [4]:
                                Numeric=8.7807862853e-01
                                Analytic=8.7741382756e-01
                                iterations=9000
Estamos em L = 27.68
Energy [0]:
                                Numeric=3.6049031396e-02
                                Analytic=3.6017084372e-02
                                iterations=14000
Energy [1]:
                                Numeric=1.4351694771e-01
                                Analytic=1.4339242258e-01
                                iterations=10000
Energy [2]:
                                Numeric=3.2002116991e-01
                                Analytic=3.1975508218e-01
                                iterations=7000
Energy [3]:
                                Numeric=5.5987849867e-01
                                Analytic=5.5943840624e-01
                                iterations=7000
Energy [4]:
                                Numeric=8.4644803277e-01
                                Analytic=8.4580128982e-01
                                iterations=8000
Estamos em L = 28.38
Energy [0]:
                                Numeric=3.4524773766e-02
                                Analytic=3.4493176330e-02
                                iterations=14000
Energy [1]:
                                Numeric=1.3748532249e-01
                                Analytic=1.3736792409e-01
                                iterations=11000
Energy [2]:
                                Numeric=3.0677529995e-01
                                Analytic=3.0651742841e-01
                                iterations=7000
Energy [3]:
                                Numeric=5.3742777231e-01
                                Analytic=5.3699913749e-01
                                iterations=7000
Energy [4]:
                                Numeric=8.1568933177e-01
                                Analytic=8.1508262046e-01
                                iterations=8000
Estamos em L = 29.09
Energy [0]:
                                Numeric=3.3093579193e-02
                                Analytic=3.3063850441e-02
                                iterations=15000
Energy [1]:
                                Numeric=1.3182723909e-01
                                Analytic=1.3171293033e-01
                                iterations=11000
Energy [2]:
                                Numeric=2.9431330009e-01
                                Analytic=2.9406961812e-01
                                iterations=8000
Energy [3]:
                                Numeric=5.1621786570e-01
                                Analytic=5.1580355070e-01
                                iterations=7000
Energy [4]:
                                Numeric=7.8605944110e-01
                                Analytic=7.8544779063e-01
                                iterations=7000
Estamos em L = 29.80
Energy [0]:
                                Numeric=3.1750989414e-02
                                Analytic=3.1721451053e-02
                                iterations=15000
Energy [1]:
                                Numeric=1.2651022724e-01
                                Analytic=1.2639807314e-01
                                iterations=11000
Energy [2]:
                                Numeric=2.8258699575e-01
                                Analytic=2.8235184108e-01
                                iterations=8000
Energy [3]:
                                Numeric=4.9617331541e-01
                                Analytic=4.9577297042e-01
                                iterations=7000
Energy [4]:
                                Numeric=7.5757186815e-01
                                Analytic=7.5698691492e-01
                                iterations=7000
Estamos em L = 30.51
Energy [0]:
                                Numeric=3.0486895326e-02
                                Analytic=3.0459080254e-02
                                iterations=16000
Energy [1]:
                                Numeric=1.2150234495e-01
                                Analytic=1.2139681506e-01
                                iterations=12000
Energy [2]:
                                Numeric=2.7153873174e-01
                                Analytic=2.7130958797e-01
                                iterations=8000
Energy [3]:
                                Numeric=4.7722491952e-01
                                Analytic=4.7683265955e-01
                                iterations=7000
Energy [4]:
                                Numeric=7.3029517181e-01
                                Analytic=7.2973211299e-01
                                iterations=7000
Energy [5]:
                                Numeric=9.9182046597e-01
                                Analytic=9.8988701840e-01
                                iterations=20000
Estamos em L = 31.21
Energy [0]:
                                Numeric=2.9298289326e-02
                                Analytic=2.9270509818e-02
                                iterations=16000
Energy [1]:
                                Numeric=1.1678854657e-01
                                Analytic=1.1668513249e-01
                                iterations=12000
Energy [2]:
                                Numeric=2.6111571174e-01
                                Analytic=2.6089313978e-01
                                iterations=8000
Energy [3]:
                                Numeric=4.5928231352e-01
                                Analytic=4.5891214420e-01
                                iterations=8000
Energy [4]:
                                Numeric=7.0422395169e-01
                                Analytic=7.0368074793e-01
                                iterations=7000
Energy [5]:
                                Numeric=9.6746384556e-01
                                Analytic=9.6647148151e-01
                                iterations=15000
Estamos em L = 31.92
Energy [0]:
                                Numeric=2.8176275003e-02
                                Analytic=2.8150104830e-02
                                iterations=17000
Energy [1]:
                                Numeric=1.1233909363e-01
                                Analytic=1.1224123869e-01
                                iterations=13000
Energy [2]:
                                Numeric=2.5126789045e-01
                                Analytic=2.5105710490e-01
                                iterations=9000
Energy [3]:
                                Numeric=4.4230641208e-01
                                Analytic=4.4194531377e-01
                                iterations=8000
Energy [4]:
                                Numeric=6.7934119465e-01
                                Analytic=6.7880911221e-01
                                iterations=7000
Energy [5]:
                                Numeric=9.4059362866e-01
                                Analytic=9.3977572157e-01
                                iterations=12000
Estamos em L = 32.63
Energy [0]:
                                Numeric=2.7117585792e-02
                                Analytic=2.7092757257e-02
                                iterations=18000
Energy [1]:
                                Numeric=1.0814121233e-01
                                Analytic=1.0804534120e-01
                                iterations=13000
Energy [2]:
                                Numeric=2.4196535570e-01
                                Analytic=2.4176000106e-01
                                iterations=9000
Energy [3]:
                                Numeric=4.2621993014e-01
                                Analytic=4.2587037705e-01
                                iterations=8000
Energy [4]:
                                Numeric=6.5559671092e-01
                                Analytic=6.5508083497e-01
                                iterations=7000
Energy [5]:
                                Numeric=9.1291173575e-01
                                Analytic=9.1214425039e-01
                                iterations=10000
Estamos em L = 33.33
Energy [0]:
                                Numeric=2.6118593760e-02
                                Analytic=2.6093828026e-02
                                iterations=18000
Energy [1]:
                                Numeric=1.0417410572e-01
                                Analytic=1.0407942900e-01
                                iterations=13000
Energy [2]:
                                Numeric=2.3316341076e-01
                                Analytic=2.3296387866e-01
                                iterations=9000
Energy [3]:
                                Numeric=4.1096815544e-01
                                Analytic=4.1062972816e-01
                                iterations=8000
Energy [4]:
                                Numeric=6.3295225116e-01
                                Analytic=6.3245220289e-01
                                iterations=7000
Energy [5]:
                                Numeric=8.8522294590e-01
                                Analytic=8.8450302434e-01
                                iterations=9000
Estamos em L = 34.04
Energy [0]:
                                Numeric=2.5172589807e-02
                                Analytic=2.5149096376e-02
                                iterations=19000
Energy [1]:
                                Numeric=1.0041634792e-01
                                Analytic=1.0032708535e-01
                                iterations=14000
Energy [2]:
                                Numeric=2.2482987681e-01
                                Analytic=2.2463398216e-01
                                iterations=9000
Energy [3]:
                                Numeric=3.9650261151e-01
                                Analytic=3.9616975846e-01
                                iterations=8000
Energy [4]:
                                Numeric=6.1136003871e-01
                                Analytic=6.1087559633e-01
                                iterations=7000
Energy [5]:
                                Numeric=8.5795723943e-01
                                Analytic=8.5730108809e-01
                                iterations=9000
Estamos em L = 34.75
Energy [0]:
                                Numeric=2.4278294743e-02
                                Analytic=2.4254715466e-02
                                iterations=19000
Energy [1]:
                                Numeric=9.6861452283e-02
                                Analytic=9.6773322857e-02
                                iterations=14000
Energy [2]:
                                Numeric=2.1692247757e-01
                                Analytic=2.1673844569e-01
                                iterations=10000
Energy [3]:
                                Numeric=3.8275460241e-01
                                Analytic=3.8244063851e-01
                                iterations=9000
Energy [4]:
                                Numeric=5.9077858500e-01
                                Analytic=5.9030173471e-01
                                iterations=7000
Energy [5]:
                                Numeric=8.3143702722e-01
                                Analytic=8.3077498245e-01
                                iterations=8000
Estamos em L = 35.45
Energy [0]:
                                Numeric=2.3429533971e-02
                                Analytic=2.3407173382e-02
                                iterations=20000
Energy [1]:
                                Numeric=9.3487854064e-02
                                Analytic=9.3404437965e-02
                                iterations=15000
Energy [2]:
                                Numeric=2.0942813744e-01
                                Analytic=2.0924801994e-01
                                iterations=10000
Energy [3]:
                                Numeric=3.6970358729e-01
                                Analytic=3.6939608633e-01
                                iterations=9000
Energy [4]:
                                Numeric=5.7114506152e-01
                                Analytic=5.7068114937e-01
                                iterations=7000
Energy [5]:
                                Numeric=8.0568020243e-01
                                Analytic=8.0505134347e-01
                                iterations=8000
Estamos em L = 36.16
Energy [0]:
                                Numeric=2.2625851733e-02
                                Analytic=2.2603258788e-02
                                iterations=20000
Energy [1]:
                                Numeric=9.0290273691e-02
                                Analytic=9.0207882209e-02
                                iterations=15000
Energy [2]:
                                Numeric=2.0231139839e-01
                                Analytic=2.0213582686e-01
                                iterations=10000
Energy [3]:
                                Numeric=3.5729153806e-01
                                Analytic=3.5699313243e-01
                                iterations=9000
Energy [4]:
                                Numeric=5.5241663342e-01
                                Analytic=5.5196514500e-01
                                iterations=7000
Energy [5]:
                                Numeric=7.8079879270e-01
                                Analytic=7.8019462235e-01
                                iterations=8000
Energy [6]:
                                Numeric=1.0007108711e+00
                                Analytic=9.9967865883e-01
                                iterations=46000
Estamos em L = 36.87
Energy [0]:
                                Numeric=2.1861445134e-02
                                Analytic=2.1840030615e-02
                                iterations=21000
Energy [1]:
                                Numeric=8.7254249977e-02
                                Analytic=8.7172148158e-02
                                iterations=15000
Energy [2]:
                                Numeric=1.9555047071e-01
                                Analytic=1.9537713943e-01
                                iterations=10000
Energy [3]:
                                Numeric=3.4548163247e-01
                                Analytic=3.4519188876e-01
                                iterations=9000
Energy [4]:
                                Numeric=5.3453184241e-01
                                Analytic=5.3410641695e-01
                                iterations=8000
Energy [5]:
                                Numeric=7.5681535033e-01
                                Analytic=7.5623203684e-01
                                iterations=8000
Energy [6]:
                                Numeric=9.8554730482e-01
                                Analytic=9.8440374092e-01
                                iterations=20000
Estamos em L = 37.58
Energy [0]:
                                Numeric=2.1135190639e-02
                                Analytic=2.1114791258e-02
                                iterations=22000
Energy [1]:
                                Numeric=8.4363986435e-02
                                Analytic=8.4286668167e-02
                                iterations=16000
Energy [2]:
                                Numeric=1.8911269720e-01
                                Analytic=1.8894918387e-01
                                iterations=11000
Energy [3]:
                                Numeric=3.3424219049e-01
                                Analytic=3.3395532590e-01
                                iterations=9000
Energy [4]:
                                Numeric=5.1747895196e-01
                                Analytic=5.1705943464e-01
                                iterations=8000
Energy [5]:
                                Numeric=7.3373180018e-01
                                Analytic=7.3316772113e-01
                                iterations=8000
Energy [6]:
                                Numeric=9.6327832643e-01
                                Analytic=9.6234527513e-01
                                iterations=14000
Estamos em L = 38.28
Energy [0]:
                                Numeric=2.0445705779e-02
                                Analytic=2.0425062814e-02
                                iterations=22000
Energy [1]:
                                Numeric=8.1618807972e-02
                                Analytic=8.1541724340e-02
                                iterations=16000
Energy [2]:
                                Numeric=1.8299051111e-01
                                Analytic=1.8283096193e-01
                                iterations=11000
Energy [3]:
                                Numeric=3.2351789784e-01
                                Analytic=3.2324906098e-01
                                iterations=10000
Energy [4]:
                                Numeric=5.0118818006e-01
                                Analytic=5.0078066477e-01
                                iterations=8000
Energy [5]:
                                Numeric=7.1154855784e-01
                                Analytic=7.1099123458e-01
                                iterations=8000
Energy [6]:
                                Numeric=9.3925793974e-01
                                Analytic=9.3847063393e-01
                                iterations=12000
Estamos em L = 38.99
Energy [0]:
                                Numeric=1.9788218243e-02
                                Analytic=1.9768565975e-02
                                iterations=23000
Energy [1]:
                                Numeric=7.9001174616e-02
                                Analytic=7.8928368318e-02
                                iterations=17000
Energy [2]:
                                Numeric=1.7716066362e-01
                                Analytic=1.7700309103e-01
                                iterations=11000
Energy [3]:
                                Numeric=3.1330610143e-01
                                Analytic=3.1304115803e-01
                                iterations=10000
Energy [4]:
                                Numeric=4.8562450564e-01
                                Analytic=4.8522868448e-01
                                iterations=8000
Energy [5]:
                                Numeric=6.9022491809e-01
                                Analytic=6.8968290423e-01
                                iterations=8000
Energy [6]:
                                Numeric=9.1487109024e-01
                                Analytic=9.1411019055e-01
                                iterations=10000
Estamos em L = 39.70
Energy [0]:
                                Numeric=1.9161984770e-02
                                Analytic=1.9143201251e-02
                                iterations=24000
Energy [1]:
                                Numeric=7.6511007915e-02
                                Analytic=7.6438349704e-02
                                iterations=17000
Energy [2]:
                                Numeric=1.7159683943e-01
                                Analytic=1.7144766043e-01
                                iterations=12000
Energy [3]:
                                Numeric=3.0355991219e-01
                                Analytic=3.0330194120e-01
                                iterations=10000
Energy [4]:
                                Numeric=4.7075572104e-01
                                Analytic=4.7036421916e-01
                                iterations=8000
Energy [5]:
                                Numeric=6.6974422829e-01
                                Analytic=6.6921728858e-01
                                iterations=8000
Energy [6]:
                                Numeric=8.9055968331e-01
                                Analytic=8.8983488476e-01
                                iterations=9000
Estamos em L = 40.40
Energy [0]:
                                Numeric=1.8566093493e-02
                                Analytic=1.8547032232e-02
                                iterations=24000
Energy [1]:
                                Numeric=7.4132830098e-02
                                Analytic=7.4064052081e-02
                                iterations=18000
Energy [2]:
                                Numeric=1.6629384660e-01
                                Analytic=1.6614810166e-01
                                iterations=12000
Energy [3]:
                                Numeric=2.9425969700e-01
                                Analytic=2.9400382094e-01
                                iterations=10000
Energy [4]:
                                Numeric=4.5653246334e-01
                                Analytic=4.5615012889e-01
                                iterations=8000
Energy [5]:
                                Numeric=6.5007741851e-01
                                Analytic=6.4956547665e-01
                                iterations=8000
Energy [6]:
                                Numeric=8.6660065964e-01
                                Analytic=8.6593483383e-01
                                iterations=9000
Estamos em L = 41.11
Energy [0]:
                                Numeric=1.7996477679e-02
                                Analytic=1.7978270632e-02
                                iterations=25000
Energy [1]:
                                Numeric=7.1867171475e-02
                                Analytic=7.1798435728e-02
                                iterations=18000
Energy [2]:
                                Numeric=1.6123316722e-01
                                Analytic=1.6108907172e-01
                                iterations=12000
Energy [3]:
                                Numeric=2.8536094491e-01
                                Analytic=2.8512113299e-01
                                iterations=11000
Energy [4]:
                                Numeric=4.4291202063e-01
                                Analytic=4.4255136060e-01
                                iterations=9000
Energy [5]:
                                Numeric=6.3120488071e-01
                                Analytic=6.3069663909e-01
                                iterations=8000
Energy [6]:
                                Numeric=8.4324692185e-01
                                Analytic=8.4256862376e-01
                                iterations=8000
Estamos em L = 41.82
Energy [0]:
                                Numeric=1.7452711297e-02
                                Analytic=1.7435262914e-02
                                iterations=26000
Energy [1]:
                                Numeric=6.9700159502e-02
                                Analytic=6.9634986246e-02
                                iterations=19000
Energy [2]:
                                Numeric=1.5639944044e-01
                                Analytic=1.5625634767e-01
                                iterations=12000
Energy [3]:
                                Numeric=2.7686690721e-01
                                Analytic=2.7662998955e-01
                                iterations=11000
Energy [4]:
                                Numeric=4.2988461669e-01
                                Analytic=4.2953487775e-01
                                iterations=9000
Energy [5]:
                                Numeric=6.1307530430e-01
                                Analytic=6.1257908567e-01
                                iterations=8000
Energy [6]:
                                Numeric=8.2047024109e-01
                                Analytic=8.1982446342e-01
                                iterations=8000
Estamos em L = 42.53
Energy [0]:
                                Numeric=1.6934251371e-02
                                Analytic=1.6916478287e-02
                                iterations=26000
Energy [1]:
                                Numeric=6.7632917205e-02
                                Analytic=6.7567668421e-02
                                iterations=19000
Energy [2]:
                                Numeric=1.5177049222e-01
                                Analytic=1.5163673141e-01
                                iterations=13000
Energy [3]:
                                Numeric=2.6873949307e-01
                                Analytic=2.6850814208e-01
                                iterations=11000
Energy [4]:
                                Numeric=4.1741653792e-01
                                Analytic=4.1706957599e-01
                                iterations=9000
Energy [5]:
                                Numeric=5.9566557600e-01
                                Analytic=5.9518098902e-01
                                iterations=8000
Energy [6]:
                                Numeric=7.9837115452e-01
                                Analytic=7.9774962537e-01
                                iterations=8000
Energy [7]:
                                Numeric=9.9863540089e-01
                                Analytic=9.9648897578e-01
                                iterations=26000
Estamos em L = 43.23
Energy [0]:
                                Numeric=1.6437520152e-02
                                Analytic=1.6420497957e-02
                                iterations=27000
Energy [1]:
                                Numeric=6.5652828861e-02
                                Analytic=6.5590884714e-02
                                iterations=20000
Energy [2]:
                                Numeric=1.4735041929e-01
                                Analytic=1.4721796358e-01
                                iterations=13000
Energy [3]:
                                Numeric=2.6096506735e-01
                                Analytic=2.6073485502e-01
                                iterations=11000
Energy [4]:
                                Numeric=4.0546445839e-01
                                Analytic=4.0512619087e-01
                                iterations=9000
Energy [5]:
                                Numeric=5.7892662078e-01
                                Analytic=5.7847087910e-01
                                iterations=9000
Energy [6]:
                                Numeric=7.7696762329e-01
                                Analytic=7.7636623138e-01
                                iterations=8000
Energy [7]:
                                Numeric=9.8057125492e-01
                                Analytic=9.7947490366e-01
                                iterations=18000
Estamos em L = 43.94
Energy [0]:
                                Numeric=1.5962354453e-02
                                Analytic=1.5946005431e-02
                                iterations=28000
Energy [1]:
                                Numeric=6.3761582961e-02
                                Analytic=6.3699437872e-02
                                iterations=20000
Energy [2]:
                                Numeric=1.4312037534e-01
                                Analytic=1.4298864562e-01
                                iterations=13000
Energy [3]:
                                Numeric=2.5350607035e-01
                                Analytic=2.5329078962e-01
                                iterations=12000
Energy [4]:
                                Numeric=3.9400719151e-01
                                Analytic=3.9367720169e-01
                                iterations=9000
Energy [5]:
                                Numeric=5.6287102485e-01
                                Analytic=5.6241797749e-01
                                iterations=9000
Energy [6]:
                                Numeric=7.5626373975e-01
                                Analytic=7.5568038998e-01
                                iterations=8000
Energy [7]:
                                Numeric=9.5986840953e-01
                                Analytic=9.5897687552e-01
                                iterations=14000
Estamos em L = 44.65
Energy [0]:
                                Numeric=1.5508506200e-02
                                Analytic=1.5491777807e-02
                                iterations=28000
Energy [1]:
                                Numeric=6.1947549085e-02
                                Analytic=6.1888497189e-02
                                iterations=21000
Energy [2]:
                                Numeric=1.3906146720e-01
                                Analytic=7.3124284724e-02
                                iterations=14000
Energy [3]:
                                Numeric=2.4637132106e-01
                                Analytic=1.3893816914e-01
                                iterations=12000
Energy [4]:
                                Numeric=3.8302616688e-01
                                Analytic=2.4615789725e-01
                                iterations=9000
Energy [5]:
                                Numeric=5.4743403625e-01
                                Analytic=3.8269673455e-01
                                iterations=9000
Energy [6]:
                                Numeric=7.3625421168e-01
                                Analytic=5.4699241886e-01
                                iterations=8000
Energy [7]:
                                Numeric=9.3823009558e-01
                                Analytic=7.3568779316e-01
                                iterations=12000
Energy [8]:
                                Numeric=1.0027410067e+00
                                Analytic=9.3743516691e-01
                                iterations=64000
Estamos em L = 45.35
Energy [0]:
                                Numeric=1.5072736001e-02
                                Analytic=1.5056677900e-02
                                iterations=29000
Energy [1]:
                                Numeric=6.0212951305e-02
                                Analytic=6.0153568030e-02
                                iterations=21000
Energy [2]:
                                Numeric=1.3517898585e-01
                                Analytic=1.3505665184e-01
                                iterations=14000
Energy [3]:
                                Numeric=2.3952837567e-01
                                Analytic=2.3931932143e-01
                                iterations=12000
Energy [4]:
                                Numeric=3.7246936587e-01
                                Analytic=3.7216046651e-01
                                iterations=10000
Energy [5]:
                                Numeric=5.3259575620e-01
                                Analytic=5.3216539209e-01
                                iterations=9000
Energy [6]:
                                Numeric=7.1694187139e-01
                                Analytic=7.1637729409e-01
                                iterations=8000
Energy [7]:
                                Numeric=9.1638131344e-01
                                Analytic=9.1565218532e-01
                                iterations=11000
Estamos em L = 46.06
Energy [0]:
                                Numeric=1.4655100060e-02
                                Analytic=1.4639647142e-02
                                iterations=30000
Energy [1]:
                                Numeric=5.8546928904e-02
                                Analytic=5.8490464255e-02
                                iterations=22000
Energy [2]:
                                Numeric=1.3145677726e-01
                                Analytic=1.3133487933e-01
                                iterations=14000
Energy [3]:
                                Numeric=2.3296822464e-01
                                Analytic=2.3275930779e-01
                                iterations=12000
Energy [4]:
                                Numeric=3.6234624345e-01
                                Analytic=3.6204553239e-01
                                iterations=10000
Energy [5]:
                                Numeric=5.1832852142e-01
                                Analytic=5.1790922411e-01
                                iterations=9000
Energy [6]:
                                Numeric=6.9828463248e-01
                                Analytic=6.9773326963e-01
                                iterations=8000
Energy [7]:
                                Numeric=8.9469999291e-01
                                Analytic=8.9399772467e-01
                                iterations=10000
Estamos em L = 46.77
Energy [0]:
                                Numeric=1.4255591197e-02
                                Analytic=1.4239699157e-02
                                iterations=30000
Energy [1]:
                                Numeric=5.6952213213e-02
                                Analytic=5.6895283235e-02
                                iterations=22000
Energy [2]:
                                Numeric=1.2787837636e-01
                                Analytic=1.2776425225e-01
                                iterations=15000
Energy [3]:
                                Numeric=2.2665768862e-01
                                Analytic=2.2646312154e-01
                                iterations=13000
Energy [4]:
                                Numeric=3.5263009146e-01
                                Analytic=3.5233043495e-01
                                iterations=10000
Energy [5]:
                                Numeric=5.0461765865e-01
                                Analytic=5.0419742271e-01
                                iterations=9000
Energy [6]:
                                Numeric=6.8027584792e-01
                                Analytic=6.7973721984e-01
                                iterations=8000
Energy [7]:
                                Numeric=8.7336440556e-01
                                Analytic=8.7266776108e-01
                                iterations=9000
Estamos em L = 47.47
Energy [0]:
                                Numeric=1.3871199984e-02
                                Analytic=1.3855913922e-02
                                iterations=31000
Energy [1]:
                                Numeric=5.5418539188e-02
                                Analytic=5.5364383194e-02
                                iterations=23000
Energy [2]:
                                Numeric=1.2445022740e-01
                                Analytic=1.2433673811e-01
                                iterations=15000
Energy [3]:
                                Numeric=2.2061070526e-01
                                Analytic=2.2041697158e-01
                                iterations=13000
Energy [4]:
                                Numeric=3.4328847067e-01
                                Analytic=3.4299495923e-01
                                iterations=10000
Energy [5]:
                                Numeric=4.9140048822e-01
                                Analytic=4.9100469001e-01
                                iterations=10000
Energy [6]:
                                Numeric=6.6289504988e-01
                                Analytic=6.6236887112e-01
                                iterations=8000
Energy [7]:
                                Numeric=8.5244250300e-01
                                Analytic=8.5177298333e-01
                                iterations=9000
Estamos em L = 48.18
Energy [0]:
                                Numeric=1.3502168463e-02
                                Analytic=1.3487432507e-02
                                iterations=32000
Energy [1]:
                                Numeric=5.3949120562e-02
                                Analytic=5.3894362618e-02
                                iterations=23000
Energy [2]:
                                Numeric=1.2115817062e-01
                                Analytic=1.2104482746e-01
                                iterations=15000
Energy [3]:
                                Numeric=2.1479834608e-01
                                Analytic=2.1460794088e-01
                                iterations=13000
Energy [4]:
                                Numeric=3.3430794553e-01
                                Analytic=3.3402009106e-01
                                iterations=10000
Energy [5]:
                                Numeric=4.7869193454e-01
                                Analytic=4.7830691519e-01
                                iterations=10000
Energy [6]:
                                Numeric=6.4612094643e-01
                                Analytic=6.4560694724e-01
                                iterations=8000
Energy [7]:
                                Numeric=8.3201856451e-01
                                Analytic=8.3137696747e-01
                                iterations=9000
Estamos em L = 48.89
Energy [0]:
                                Numeric=1.3147685559e-02
                                Analytic=1.3133452252e-02
                                iterations=33000
Energy [1]:
                                Numeric=5.2534145435e-02
                                Analytic=5.2482041542e-02
                                iterations=24000
Energy [2]:
                                Numeric=1.1798853920e-01
                                Analytic=1.1788149388e-01
                                iterations=16000
Energy [3]:
                                Numeric=2.0920489574e-01
                                Analytic=2.0902392252e-01
                                iterations=14000
Energy [4]:
                                Numeric=3.2566252369e-01
                                Analytic=3.2538794027e-01
                                iterations=11000
Energy [5]:
                                Numeric=4.6646568837e-01
                                Analytic=4.6608115259e-01
                                iterations=10000
Energy [6]:
                                Numeric=6.2992535276e-01
                                Analytic=6.2942971292e-01
                                iterations=9000
Energy [7]:
                                Numeric=8.1213512907e-01
                                Analytic=8.1151518369e-01
                                iterations=9000
Energy [8]:
                                Numeric=9.9379612289e-01
                                Analytic=9.9220231269e-01
                                iterations=22000
Estamos em L = 49.60
Energy [0]:
                                Numeric=1.2807907194e-02
                                Analytic=1.2793222420e-02
                                iterations=33000
Energy [1]:
                                Numeric=5.1177289210e-02
                                Analytic=5.1124444499e-02
                                iterations=24000
Energy [2]:
                                Numeric=1.1494589093e-01
                                Analytic=1.1484015749e-01
                                iterations=16000
Energy [3]:
                                Numeric=2.0383066272e-01
                                Analytic=2.0365356087e-01
                                iterations=14000
Energy [4]:
                                Numeric=3.1735002466e-01
                                Analytic=3.1708166817e-01
                                iterations=11000
Energy [5]:
                                Numeric=4.5468176850e-01
                                Analytic=4.5430558976e-01
                                iterations=10000
Energy [6]:
                                Numeric=6.1429935812e-01
                                Analytic=6.1381535854e-01
                                iterations=9000
Energy [7]:
                                Numeric=7.9280687796e-01
                                Analytic=7.9220544068e-01
                                iterations=9000
Energy [8]:
                                Numeric=9.7622249635e-01
                                Analytic=9.7517443620e-01
                                iterations=16000
Estamos em L = 50.30
Energy [0]:
                                Numeric=1.2480220160e-02
                                Analytic=1.2466040204e-02
                                iterations=34000
Energy [1]:
                                Numeric=4.9869074855e-02
                                Analytic=4.9818784981e-02
                                iterations=25000
Energy [2]:
                                Numeric=1.1202052285e-01
                                Analytic=1.1191465157e-01
                                iterations=16000
Energy [3]:
                                Numeric=1.9866346477e-01
                                Analytic=1.3006329597e-01
                                iterations=14000
Energy [4]:
                                Numeric=3.0934787926e-01
                                Analytic=1.9848619755e-01
                                iterations=11000
Energy [5]:
                                Numeric=4.4332771132e-01
                                Analytic=3.0908541971e-01
                                iterations=10000
Energy [6]:
                                Numeric=5.9921474507e-01
                                Analytic=4.4295950883e-01
                                iterations=9000
Energy [7]:
                                Numeric=7.7403857789e-01
                                Analytic=5.9874227213e-01
                                iterations=9000
Energy [8]:
                                Numeric=9.5707324477e-01
                                Analytic=7.7345405080e-01
                                iterations=13000
Energy [9]:
                                Numeric=1.0144712013e+00
                                Analytic=9.5619055627e-01
                                iterations=47000
Estamos em L = 51.01
Energy [0]:
                                Numeric=1.2164964307e-02
                                Analytic=1.2151247119e-02
                                iterations=35000
Energy [1]:
                                Numeric=4.8613622891e-02
                                Analytic=4.8562451255e-02
                                iterations=25000
Energy [2]:
                                Numeric=1.0919914568e-01
                                Analytic=1.0909919204e-01
                                iterations=17000
Energy [3]:
                                Numeric=1.9368668396e-01
                                Analytic=1.9351182174e-01
                                iterations=14000
Energy [4]:
                                Numeric=3.0164763414e-01
                                Analytic=3.0138425968e-01
                                iterations=11000
Energy [5]:
                                Numeric=4.3238381088e-01
                                Analytic=4.3202324362e-01
                                iterations=10000
Energy [6]:
                                Numeric=5.8465035929e-01
                                Analytic=5.8418923036e-01
                                iterations=9000
Energy [7]:
                                Numeric=7.5582802654e-01
                                Analytic=7.5525967045e-01
                                iterations=9000
Energy [8]:
                                Numeric=9.3741567436e-01
                                Analytic=9.3659421702e-01
                                iterations=11000
Estamos em L = 51.72
Energy [0]:
                                Numeric=1.1861516569e-02
                                Analytic=1.1848225685e-02
                                iterations=36000
Energy [1]:
                                Numeric=4.7401691819e-02
                                Analytic=4.7352993405e-02
                                iterations=26000
Energy [2]:
                                Numeric=1.0648819545e-01
                                Analytic=1.0638834954e-01
                                iterations=17000
Energy [3]:
                                Numeric=1.8888646785e-01
                                Analytic=1.8872102443e-01
                                iterations=15000
Energy [4]:
                                Numeric=2.9421077241e-01
                                Analytic=2.9396411327e-01
                                iterations=12000
Energy [5]:
                                Numeric=4.2182445013e-01
                                Analytic=4.2147813430e-01
                                iterations=11000
Energy [6]:
                                Numeric=5.7059800489e-01
                                Analytic=5.7013553074e-01
                                iterations=9000
Energy [7]:
                                Numeric=7.3818637601e-01
                                Analytic=7.3761579347e-01
                                iterations=9000
Energy [8]:
                                Numeric=9.1765994963e-01
                                Analytic=9.1689479954e-01
                                iterations=10000
Estamos em L = 52.42
Energy [0]:
                                Numeric=1.1569292863e-02
                                Analytic=1.1556396401e-02
                                iterations=37000
Energy [1]:
                                Numeric=4.6234681541e-02
                                Analytic=4.6188111470e-02
                                iterations=27000
Energy [2]:
                                Numeric=1.0387720712e-01
                                Analytic=1.0377702378e-01
                                iterations=17000
Energy [3]:
                                Numeric=1.8427051066e-01
                                Analytic=1.8410495625e-01
                                iterations=15000
Energy [4]:
                                Numeric=2.8705247446e-01
                                Analytic=2.8681171039e-01
                                iterations=12000
Energy [5]:
                                Numeric=4.1164473699e-01
                                Analytic=4.1130648095e-01
                                iterations=11000
Energy [6]:
                                Numeric=5.5701477558e-01
                                Analytic=5.5656108078e-01
                                iterations=9000
Energy [7]:
                                Numeric=7.2107063738e-01
                                Analytic=7.2051242290e-01
                                iterations=9000
Energy [8]:
                                Numeric=8.9803150966e-01
                                Analytic=8.9734169566e-01
                                iterations=10000
Estamos em L = 53.13
Energy [0]:
                                Numeric=1.1288556740e-02
                                Analytic=1.1275214972e-02
                                iterations=37000
Energy [1]:
                                Numeric=4.5112961043e-02
                                Analytic=4.5065644592e-02
                                iterations=27000
Energy [2]:
                                Numeric=1.0135412156e-01
                                Analytic=1.0126042007e-01
                                iterations=18000
Energy [3]:
                                Numeric=1.7981885331e-01
                                Analytic=1.7965528872e-01
                                iterations=15000
Energy [4]:
                                Numeric=2.8015603769e-01
                                Analytic=2.7991453380e-01
                                iterations=12000
Energy [5]:
                                Numeric=4.0182187266e-01
                                Analytic=4.0149149701e-01
                                iterations=11000
Energy [6]:
                                Numeric=5.4389184874e-01
                                Analytic=5.4344645549e-01
                                iterations=9000
Energy [7]:
                                Numeric=7.0448337257e-01
                                Analytic=7.0393722059e-01
                                iterations=9000
Energy [8]:
                                Numeric=8.7876143448e-01
                                Analytic=8.7807210253e-01
                                iterations=9000
Estamos em L = 53.84
Energy [0]:
                                Numeric=1.1017115364e-02
                                Analytic=1.1004169787e-02
                                iterations=38000
Energy [1]:
                                Numeric=4.4028799049e-02
                                Analytic=4.3983561060e-02
                                iterations=28000
Energy [2]:
                                Numeric=9.8927913078e-02
                                Analytic=9.8834027723e-02
                                iterations=18000
Energy [3]:
                                Numeric=1.7551911156e-01
                                Analytic=1.7536417843e-01
                                iterations=16000
Energy [4]:
                                Numeric=2.7349800850e-01
                                Analytic=2.7326077073e-01
                                iterations=12000
Energy [5]:
                                Numeric=3.9235016639e-01
                                Analytic=3.9201726326e-01
                                iterations=11000
Energy [6]:
                                Numeric=5.3119070850e-01
                                Analytic=5.3077293155e-01
                                iterations=10000
Energy [7]:
                                Numeric=6.8841069622e-01
                                Analytic=6.8787631323e-01
                                iterations=9000
Energy [8]:
                                Numeric=8.5983397394e-01
                                Analytic=8.5916578781e-01
                                iterations=9000
Estamos em L = 54.55
Energy [0]:
                                Numeric=1.0755358152e-02
                                Analytic=1.0742779590e-02
                                iterations=39000
Energy [1]:
                                Numeric=4.2986082175e-02
                                Analytic=4.2939949176e-02
                                iterations=28000
Energy [2]:
                                Numeric=9.6588093785e-02
                                Analytic=9.6493600231e-02
                                iterations=18000
Energy [3]:
                                Numeric=1.7137664463e-01
                                Analytic=1.7122423401e-01
                                iterations=16000
Energy [4]:
                                Numeric=2.6707260935e-01
                                Analytic=2.6683926779e-01
                                iterations=12000
Energy [5]:
                                Numeric=3.8319618003e-01
                                Analytic=3.8286868283e-01
                                iterations=11000
Energy [6]:
                                Numeric=5.1892960608e-01
                                Analytic=5.1852250418e-01
                                iterations=10000
Energy [7]:
                                Numeric=6.7283758065e-01
                                Analytic=6.7231486616e-01
                                iterations=9000
Energy [8]:
                                Numeric=8.4131152188e-01
                                Analytic=8.4066984642e-01
                                iterations=9000
Energy [9]:
                                Numeric=1.0031932322e+00
                                Analytic=9.9978391933e-01
                                iterations=31000
Estamos em L = 55.25
Energy [0]:
                                Numeric=1.0502828776e-02
                                Analytic=1.0490591333e-02
                                iterations=40000
Energy [1]:
                                Numeric=4.1977102802e-02
                                Analytic=4.1933008862e-02
                                iterations=29000
Energy [2]:
                                Numeric=9.4324166985e-02
                                Analytic=9.4235136999e-02
                                iterations=19000
Energy [3]:
                                Numeric=1.6738197242e-01
                                Analytic=1.6722848570e-01
                                iterations=16000
Energy [4]:
                                Numeric=2.6086199134e-01
                                Analytic=2.6063948898e-01
                                iterations=13000
Energy [5]:
                                Numeric=3.7433622144e-01
                                Analytic=3.7403143765e-01
                                iterations=12000
Energy [6]:
                                Numeric=5.0708809197e-01
                                Analytic=5.0667789137e-01
                                iterations=10000
Energy [7]:
                                Numeric=6.5776626920e-01
                                Analytic=6.5723749615e-01
                                iterations=9000
Energy [8]:
                                Numeric=8.2323249244e-01
                                Analytic=8.2261142535e-01
                                iterations=9000
Energy [9]:
                                Numeric=9.8919271889e-01
                                Analytic=9.8788806076e-01
                                iterations=20000
Estamos em L = 55.96
Energy [0]:
                                Numeric=1.0259097680e-02
                                Analytic=1.0247178216e-02
                                iterations=41000
Energy [1]:
                                Numeric=4.1003347803e-02
                                Analytic=4.0961043946e-02
                                iterations=30000
Energy [2]:
                                Numeric=9.2144302915e-02
                                Analytic=9.2054866554e-02
                                iterations=19000
Energy [3]:
                                Numeric=1.6351583560e-01
                                Analytic=1.6337035716e-01
                                iterations=17000
Energy [4]:
                                Numeric=2.5486957926e-01
                                Analytic=2.5465147655e-01
                                iterations=13000
Energy [5]:
                                Numeric=3.6579854992e-01
                                Analytic=3.6549194646e-01
                                iterations=12000
Energy [6]:
                                Numeric=4.9562450939e-01
                                Analytic=4.9522252872e-01
                                iterations=10000
Energy [7]:
                                Numeric=6.4312389358e-01
                                Analytic=6.4262857055e-01
                                iterations=10000
Energy [8]:
                                Numeric=8.0560844260e-01
                                Analytic=8.0500488198e-01
                                iterations=9000
Energy [9]:
                                Numeric=9.7246993834e-01
                                Analytic=9.7145189223e-01
                                iterations=15000
Estamos em L = 56.67
Energy [0]:
                                Numeric=1.0024521545e-02
                                Analytic=1.0012137912e-02
                                iterations=41000
Energy [1]:
                                Numeric=4.0065584918e-02
                                Analytic=4.0022455061e-02
                                iterations=30000
Energy [2]:
                                Numeric=9.0033136556e-02
                                Analytic=8.9949231047e-02
                                iterations=20000
Energy [3]:
                                Numeric=1.5978702413e-01
                                Analytic=1.0281106176e-01
                                iterations=17000
Energy [4]:
                                Numeric=2.4908528241e-01
                                Analytic=1.5964363948e-01
                                iterations=13000
Energy [5]:
                                Numeric=3.5753813940e-01
                                Analytic=2.4886581459e-01
                                iterations=12000
Energy [6]:
                                Numeric=4.8453468067e-01
                                Analytic=3.5723732466e-01
                                iterations=10000
Energy [7]:
                                Numeric=6.2895672009e-01
                                Analytic=4.8414055754e-01
                                iterations=10000
Energy [8]:
                                Numeric=7.8844392264e-01
                                Analytic=6.2847242482e-01
                                iterations=9000
Energy [9]:
                                Numeric=9.5471018262e-01
                                Analytic=7.8785608853e-01
                                iterations=13000
Energy [10]:
                                Numeric=1.0020270551e+00
                                Analytic=9.5385514478e-01
                                iterations=83000
Estamos em L = 57.37
Energy [0]:
                                Numeric=9.7971550923e-03
                                Analytic=9.7850908701e-03
                                iterations=42000
Energy [1]:
                                Numeric=3.9157095845e-02
                                Analytic=3.9115733095e-02
                                iterations=31000
Energy [2]:
                                Numeric=8.7999088944e-02
                                Analytic=8.7914871970e-02
                                iterations=20000
Energy [3]:
                                Numeric=1.5618696245e-01
                                Analytic=1.5604246708e-01
                                iterations=17000
Energy [4]:
                                Numeric=2.4349015688e-01
                                Analytic=2.4327359511e-01
                                iterations=13000
Energy [5]:
                                Numeric=3.4955070084e-01
                                Analytic=3.4925534594e-01
                                iterations=12000
Energy [6]:
                                Numeric=4.7380330614e-01
                                Analytic=4.7341680816e-01
                                iterations=10000
Energy [7]:
                                Numeric=6.1522680711e-01
                                Analytic=6.1475352053e-01
                                iterations=10000
Energy [8]:
                                Numeric=7.7173825995e-01
                                Analytic=7.7116515567e-01
                                iterations=9000
Energy [9]:
                                Numeric=9.3665585832e-01
                                Analytic=9.3589801321e-01
                                iterations=12000
Estamos em L = 58.08
Energy [0]:
                                Numeric=9.5774452768e-03
                                Analytic=9.5656788045e-03
                                iterations=43000
Energy [1]:
                                Numeric=3.8281792153e-02
                                Analytic=3.8239453166e-02
                                iterations=31000
Energy [2]:
                                Numeric=8.6033508148e-02
                                Analytic=8.5948616957e-02
                                iterations=20000
Energy [3]:
                                Numeric=1.5269574629e-01
                                Analytic=1.5256129545e-01
                                iterations=18000
Energy [4]:
                                Numeric=2.3806758437e-01
                                Analytic=2.3786638642e-01
                                iterations=14000
Energy [5]:
                                Numeric=3.4183362085e-01
                                Analytic=3.4153440579e-01
                                iterations=12000
Energy [6]:
                                Numeric=4.6342859001e-01
                                Analytic=4.6303677978e-01
                                iterations=10000
Energy [7]:
                                Numeric=6.0191893546e-01
                                Analytic=6.0145655972e-01
                                iterations=10000
Energy [8]:
                                Numeric=7.5548735169e-01
                                Analytic=7.5492822579e-01
                                iterations=9000
Energy [9]:
                                Numeric=9.1863588544e-01
                                Analytic=9.1791793682e-01
                                iterations=11000
Estamos em L = 58.79
Energy [0]:
                                Numeric=9.3650515556e-03
                                Analytic=9.3535632688e-03
                                iterations=44000
Energy [1]:
                                Numeric=3.7432856699e-02
                                Analytic=3.7392269044e-02
                                iterations=32000
Energy [2]:
                                Numeric=8.4127678837e-02
                                Analytic=8.4047467597e-02
                                iterations=21000
Energy [3]:
                                Numeric=1.4933014575e-01
                                Analytic=1.4919488048e-01
                                iterations=18000
Energy [4]:
                                Numeric=2.3283882365e-01
                                Analytic=2.3263620372e-01
                                iterations=14000
Energy [5]:
                                Numeric=3.3434145174e-01
                                Analytic=3.3406348689e-01
                                iterations=13000
Energy [6]:
                                Numeric=4.5335188612e-01
                                Analytic=4.5298661810e-01
                                iterations=11000
Energy [7]:
                                Numeric=5.8903591419e-01
                                Analytic=5.8856656656e-01
                                iterations=10000
Energy [8]:
                                Numeric=7.3970535110e-01
                                Analytic=7.3913869126e-01
                                iterations=9000
Energy [9]:
                                Numeric=9.0079672948e-01
                                Analytic=9.0008885656e-01
                                iterations=10000
Estamos em L = 59.49
Energy [0]:
                                Numeric=9.1596522209e-03
                                Analytic=9.1484243520e-03
                                iterations=45000
Energy [1]:
                                Numeric=3.6611936673e-02
                                Analytic=3.6572908011e-02
                                iterations=33000
Energy [2]:
                                Numeric=8.2289334475e-02
                                Analytic=8.2208588159e-02
                                iterations=21000
Energy [3]:
                                Numeric=1.4607473161e-01
                                Analytic=1.4593825940e-01
                                iterations=18000
Energy [4]:
                                Numeric=2.2777496175e-01
                                Analytic=2.2757548167e-01
                                iterations=14000
Energy [5]:
                                Numeric=3.2710440133e-01
                                Analytic=3.2683212633e-01
                                iterations=13000
Energy [6]:
                                Numeric=4.4361068844e-01
                                Analytic=4.4325309154e-01
                                iterations=11000
Energy [7]:
                                Numeric=5.7653034647e-01
                                Analytic=5.7606894473e-01
                                iterations=10000
Energy [8]:
                                Numeric=7.2434423931e-01
                                Analytic=7.2378804320e-01
                                iterations=9000
Energy [9]:
                                Numeric=8.8316885853e-01
                                Analytic=8.8250967985e-01
                                iterations=10000
Estamos em L = 60.20
Energy [0]:
                                Numeric=8.9609431456e-03
                                Analytic=8.9499595177e-03
                                iterations=46000
Energy [1]:
                                Numeric=3.5820139720e-02
                                Analytic=3.5780166099e-02
                                iterations=33000
Energy [2]:
                                Numeric=8.0510934009e-02
                                Analytic=8.0429295158e-02
                                iterations=21000
Energy [3]:
                                Numeric=1.4291375719e-01
                                Analytic=1.4278673297e-01
                                iterations=19000
Energy [4]:
                                Numeric=2.2287367666e-01
                                Analytic=2.2267704883e-01
                                iterations=14000
Energy [5]:
                                Numeric=3.2010587075e-01
                                Analytic=3.1983038466e-01
                                iterations=13000
Energy [6]:
                                Numeric=4.3417367839e-01
                                Analytic=4.3382356665e-01
                                iterations=11000
Energy [7]:
                                Numeric=5.6440338647e-01
                                Analytic=5.6394951640e-01
                                iterations=10000
Energy [8]:
                                Numeric=7.0941259158e-01
                                Analytic=7.0886647458e-01
                                iterations=9000
Energy [9]:
                                Numeric=8.6586491698e-01
                                Analytic=8.6523941220e-01
                                iterations=10000
Estamos em L = 60.91
Energy [0]:
                                Numeric=8.7686366284e-03
                                Analytic=8.7578823942e-03
                                iterations=47000
Energy [1]:
                                Numeric=3.5051324811e-02
                                Analytic=3.5012903696e-02
                                iterations=34000
Energy [2]:
                                Numeric=7.8784046842e-02
                                Analytic=7.8707047684e-02
                                iterations=22000
Energy [3]:
                                Numeric=1.3986382164e-01
                                Analytic=1.3973584903e-01
                                iterations=19000
Energy [4]:
                                Numeric=2.1812171373e-01
                                Analytic=2.1793410385e-01
                                iterations=15000
Energy [5]:
                                Numeric=3.1332055574e-01
                                Analytic=3.1304881657e-01
                                iterations=13000
Energy [6]:
                                Numeric=4.2504101511e-01
                                Analytic=4.2468598316e-01
                                iterations=11000
Energy [7]:
                                Numeric=5.5261650417e-01
                                Analytic=5.5219454739e-01
                                iterations=11000
Energy [8]:
                                Numeric=6.9489984965e-01
                                Analytic=6.9436331563e-01
                                iterations=9000
Energy [9]:
                                Numeric=8.4893762193e-01
                                Analytic=8.4831373553e-01
                                iterations=10000
Energy [10]:
                                Numeric=9.9930815763e-01
                                Analytic=9.9717234234e-01
                                iterations=25000
Estamos em L = 61.62
Energy [0]:
                                Numeric=8.5831408514e-03
                                Analytic=8.5719218480e-03
                                iterations=47000
Energy [1]:
                                Numeric=3.4309558034e-02
                                Analytic=3.4270041453e-02
                                iterations=34000
Energy [2]:
                                Numeric=7.7116536426e-02
                                Analytic=7.7039438445e-02
                                iterations=22000
Energy [3]:
                                Numeric=1.3690823510e-01
                                Analytic=1.3678138724e-01
                                iterations=19000
Energy [4]:
                                Numeric=2.1352451247e-01
                                Analytic=2.1334019326e-01
                                iterations=15000
Energy [5]:
                                Numeric=3.0673053409e-01
                                Analytic=3.0647844338e-01
                                iterations=14000
Energy [6]:
                                Numeric=4.1617867132e-01
                                Analytic=4.1582882921e-01
                                iterations=11000
Energy [7]:
                                Numeric=5.4121838093e-01
                                Analytic=5.4079076172e-01
                                iterations=11000
Energy [8]:
                                Numeric=6.8076625278e-01
                                Analytic=6.8026735196e-01
                                iterations=10000
Energy [9]:
                                Numeric=8.3235949358e-01
                                Analytic=8.3175378638e-01
                                iterations=10000
Energy [10]:
                                Numeric=9.8503187998e-01
                                Analytic=9.8386767731e-01
                                iterations=18000
Estamos em L = 62.32
Energy [0]:
                                Numeric=8.4028107841e-03
                                Analytic=8.3918209413e-03
                                iterations=48000
Energy [1]:
                                Numeric=3.3588521671e-02
                                Analytic=3.3550556520e-02
                                iterations=35000
Energy [2]:
                                Numeric=7.5497133165e-02
                                Analytic=3.7859515977e-02
                                iterations=23000
Energy [3]:
                                Numeric=1.3403966672e-01
                                Analytic=7.5424185450e-02
                                iterations=20000
Energy [4]:
                                Numeric=2.0907565235e-01
                                Analytic=1.3391934482e-01
                                iterations=15000
Energy [5]:
                                Numeric=3.0036572943e-01
                                Analytic=2.0888919071e-01
                                iterations=14000
Energy [6]:
                                Numeric=4.0756707995e-01
                                Analytic=3.0011072695e-01
                                iterations=12000
Energy [7]:
                                Numeric=5.3014512305e-01
                                Analytic=4.0724111673e-01
                                iterations=11000
Energy [8]:
                                Numeric=6.6707291553e-01
                                Analytic=5.2972534830e-01
                                iterations=10000
Energy [9]:
                                Numeric=8.1616108290e-01
                                Analytic=6.6656705912e-01
                                iterations=10000
Energy [10]:
                                Numeric=9.6918917905e-01
                                Analytic=8.1557120455e-01
                                iterations=14000
Energy [11]:
                                Numeric=1.0103047954e+00
                                Analytic=9.6822712026e-01
                                iterations=58000
Estamos em L = 63.03
Energy [0]:
                                Numeric=8.2281110283e-03
                                Analytic=8.2173361118e-03
                                iterations=49000
Energy [1]:
                                Numeric=3.2890049052e-02
                                Analytic=3.2853479027e-02
                                iterations=36000
Energy [2]:
                                Numeric=7.3932760664e-02
                                Analytic=7.3859124294e-02
                                iterations=23000
Energy [3]:
                                Numeric=1.3126735512e-01
                                Analytic=1.3114592339e-01
                                iterations=20000
Energy [4]:
                                Numeric=2.0475969631e-01
                                Analytic=2.0457527771e-01
                                iterations=15000
Energy [5]:
                                Numeric=2.9418845552e-01
                                Analytic=2.9393754527e-01
                                iterations=14000
Energy [6]:
                                Numeric=3.9923112392e-01
                                Analytic=3.9891235751e-01
                                iterations=12000
Energy [7]:
                                Numeric=5.1939806622e-01
                                Analytic=5.1898596153e-01
                                iterations=11000
Energy [8]:
                                Numeric=6.5374725521e-01
                                Analytic=6.5325077642e-01
                                iterations=10000
Energy [9]:
                                Numeric=8.0034663056e-01
                                Analytic=7.9977122264e-01
                                iterations=10000
Energy [10]:
                                Numeric=9.5272936639e-01
                                Analytic=9.5187349981e-01
                                iterations=12000
Estamos em L = 63.74
Energy [0]:
                                Numeric=8.0588094328e-03
                                Analytic=8.0482362994e-03
                                iterations=50000
Energy [1]:
                                Numeric=3.2215546749e-02
                                Analytic=3.2177888836e-02
                                iterations=36000
Energy [2]:
                                Numeric=7.2416844106e-02
                                Analytic=7.2342200982e-02
                                iterations=23000
Energy [3]:
                                Numeric=1.2857812240e-01
                                Analytic=1.2845751659e-01
                                iterations=20000
Energy [4]:
                                Numeric=2.0056361198e-01
                                Analytic=2.0039292547e-01
                                iterations=16000
Energy [5]:
                                Numeric=2.8819822814e-01
                                Analytic=2.8795116936e-01
                                iterations=14000
Energy [6]:
                                Numeric=3.9115607847e-01
                                Analytic=3.9083253978e-01
                                iterations=12000
Energy [7]:
                                Numeric=5.0896537278e-01
                                Analytic=5.0856071750e-01
                                iterations=11000
Energy [8]:
                                Numeric=6.4079423068e-01
                                Analytic=6.4030683627e-01
                                iterations=10000
Energy [9]:
                                Numeric=7.8491633294e-01
                                Analytic=7.8435464969e-01
                                iterations=10000
Energy [10]:
                                Numeric=9.3608919136e-01
                                Analytic=9.3531233492e-01
                                iterations=11000
Estamos em L = 64.44
Energy [0]:
                                Numeric=7.8946857343e-03
                                Analytic=7.8843022249e-03
                                iterations=51000
Energy [1]:
                                Numeric=3.1559170813e-02
                                Analytic=3.1522912511e-02
                                iterations=37000
Energy [2]:
                                Numeric=7.0941945170e-02
                                Analytic=7.0871465259e-02
                                iterations=24000
Energy [3]:
                                Numeric=1.2596498494e-01
                                Analytic=1.2585069873e-01
                                iterations=21000
Energy [4]:
                                Numeric=1.9650980156e-01
                                Analytic=1.9633687813e-01
                                iterations=16000
Energy [5]:
                                Numeric=2.8238114488e-01
                                Analytic=2.8214424157e-01
                                iterations=15000
Energy [6]:
                                Numeric=3.8331049101e-01
                                Analytic=3.8299210565e-01
                                iterations=12000
Energy [7]:
                                Numeric=4.9885054300e-01
                                Analytic=4.9843818663e-01
                                iterations=11000
Energy [8]:
                                Numeric=6.2820202146e-01
                                Analytic=6.2772366028e-01
                                iterations=10000
Energy [9]:
                                Numeric=7.6986772896e-01
                                Analytic=7.6931919411e-01
                                iterations=10000
Energy [10]:
                                Numeric=9.1946257493e-01
                                Analytic=9.1877527813e-01
                                iterations=11000
Estamos em L = 65.15
Energy [0]:
                                Numeric=7.7355308288e-03
                                Analytic=7.7253256186e-03
                                iterations=52000
Energy [1]:
                                Numeric=3.0922710865e-02
                                Analytic=3.0887720511e-02
                                iterations=38000
Energy [2]:
                                Numeric=6.9516449755e-02
                                Analytic=6.9445064407e-02
                                iterations=24000
Energy [3]:
                                Numeric=1.2343778869e-01
                                Analytic=1.2332221401e-01
                                iterations=21000
Energy [4]:
                                Numeric=1.9257280572e-01
                                Analytic=1.9240213696e-01
                                iterations=16000
Energy [5]:
                                Numeric=2.7674238365e-01
                                Analytic=2.7650975520e-01
                                iterations=15000
Energy [6]:
                                Numeric=3.7569538438e-01
                                Analytic=3.7538192928e-01
                                iterations=12000
Energy [7]:
                                Numeric=4.8899151337e-01
                                Analytic=4.8860738407e-01
                                iterations=12000
Energy [8]:
                                Numeric=6.1595931609e-01
                                Analytic=6.1548983035e-01
                                iterations=10000
Energy [9]:
                                Numeric=7.5521908691e-01
                                Analytic=7.5466037318e-01
                                iterations=10000
Energy [10]:
                                Numeric=9.0307099889e-01
                                Analytic=9.0238712956e-01
                                iterations=10000
Estamos em L = 65.86
Energy [0]:
                                Numeric=7.5811460944e-03
                                Analytic=7.5711086548e-03
                                iterations=53000
Energy [1]:
                                Numeric=3.0307613317e-02
                                Analytic=3.0271524564e-02
                                iterations=38000
Energy [2]:
                                Numeric=6.8128823641e-02
                                Analytic=6.8061237468e-02
                                iterations=25000
Energy [3]:
                                Numeric=1.2098398851e-01
                                Analytic=1.2086896665e-01
                                iterations=21000
Energy [4]:
                                Numeric=1.8875751830e-01
                                Analytic=1.8858394563e-01
                                iterations=16000
Energy [5]:
                                Numeric=2.7126953597e-01
                                Analytic=2.7104103528e-01
                                iterations=15000
Energy [6]:
                                Numeric=3.6830220503e-01
                                Analytic=3.6799329600e-01
                                iterations=12000
Energy [7]:
                                Numeric=4.7943437125e-01
                                Analytic=4.7905775808e-01
                                iterations=12000
Energy [8]:
                                Numeric=6.0405490723e-01
                                Analytic=6.0359414094e-01
                                iterations=10000
Energy [9]:
                                Numeric=7.4092107049e-01
                                Analytic=7.4037215376e-01
                                iterations=10000
Energy [10]:
                                Numeric=8.8686101995e-01
                                Analytic=8.8622096638e-01
                                iterations=10000
Estamos em L = 66.57
Energy [0]:
                                Numeric=7.4313427632e-03
                                Analytic=7.4214632412e-03
                                iterations=54000
Energy [1]:
                                Numeric=2.9708387628e-02
                                Analytic=2.9673575235e-02
                                iterations=39000
Energy [2]:
                                Numeric=6.6786725397e-02
                                Analytic=6.6718309855e-02
                                iterations=25000
Energy [3]:
                                Numeric=1.1859687443e-01
                                Analytic=1.1848801157e-01
                                iterations=22000
Energy [4]:
                                Numeric=1.8503842918e-01
                                Analytic=1.8487777646e-01
                                iterations=17000
Energy [5]:
                                Numeric=2.6596470009e-01
                                Analytic=2.6573172052e-01
                                iterations=15000
Energy [6]:
                                Numeric=3.6111394006e-01
                                Analytic=3.6081788226e-01
                                iterations=13000
Energy [7]:
                                Numeric=4.7014849810e-01
                                Analytic=4.6977917737e-01
                                iterations=12000
Energy [8]:
                                Numeric=5.9249785884e-01
                                Analytic=5.9202563673e-01
                                iterations=10000
Energy [9]:
                                Numeric=7.2698682385e-01
                                Analytic=7.2644741288e-01
                                iterations=10000
Energy [10]:
                                Numeric=8.7093066216e-01
                                Analytic=8.7032139801e-01
                                iterations=10000
Estamos em L = 67.27
Energy [0]:
                                Numeric=7.2859413371e-03
                                Analytic=7.2762104514e-03
                                iterations=55000
Energy [1]:
                                Numeric=2.9126808656e-02
                                Analytic=2.9093159661e-02
                                iterations=40000
Energy [2]:
                                Numeric=6.5479609846e-02
                                Analytic=6.5414688336e-02
                                iterations=26000
Energy [3]:
                                Numeric=1.1628688657e-01
                                Analytic=1.1617654574e-01
                                iterations=22000
Energy [4]:
                                Numeric=1.8144210531e-01
                                Analytic=1.8127931750e-01
                                iterations=17000
Energy [5]:
                                Numeric=2.6079202503e-01
                                Analytic=2.6057574631e-01
                                iterations=16000
Energy [6]:
                                Numeric=3.5413873366e-01
                                Analytic=3.5384773646e-01
                                iterations=13000
Energy [7]:
                                Numeric=4.6113811008e-01
                                Analytic=4.6076191743e-01
                                iterations=12000
Energy [8]:
                                Numeric=5.8121283955e-01
                                Analytic=5.8077363932e-01
                                iterations=11000
Energy [9]:
                                Numeric=7.1340857639e-01
                                Analytic=7.1287827407e-01
                                iterations=10000
Energy [10]:
                                Numeric=8.5532779426e-01
                                Analytic=8.5471597606e-01
                                iterations=10000
Energy [11]:
                                Numeric=9.9528651477e-01
                                Analytic=9.9367085627e-01
                                iterations=22000
Estamos em L = 67.98
Energy [0]:
                                Numeric=7.1447710440e-03
                                Analytic=7.1351800886e-03
                                iterations=56000
Energy [1]:
                                Numeric=2.8564370809e-02
                                Analytic=2.8529599416e-02
                                iterations=40000
Energy [2]:
                                Numeric=6.4214550689e-02
                                Analytic=6.4148856344e-02
                                iterations=26000
Energy [3]:
                                Numeric=1.1404194022e-01
                                Analytic=1.1393190008e-01
                                iterations=22000
Energy [4]:
                                Numeric=1.7794566289e-01
                                Analytic=1.7778446051e-01
                                iterations=17000
Energy [5]:
                                Numeric=2.5577931957e-01
                                Analytic=2.5556732877e-01
                                iterations=16000
Energy [6]:
                                Numeric=3.4736133798e-01
                                Analytic=3.4707526064e-01
                                iterations=13000
Energy [7]:
                                Numeric=4.5236783699e-01
                                Analytic=4.5199664648e-01
                                iterations=12000
Energy [8]:
                                Numeric=5.7025842237e-01
                                Analytic=5.6982776526e-01
                                iterations=11000
Energy [9]:
                                Numeric=7.0017764866e-01
                                Analytic=6.9965635609e-01
                                iterations=10000
Energy [10]:
                                Numeric=8.4001606887e-01
                                Analytic=8.3942140475e-01
                                iterations=10000
Energy [11]:
                                Numeric=9.8125813956e-01
                                Analytic=9.8022271434e-01
                                iterations=17000
Estamos em L = 68.69
Energy [0]:
                                Numeric=7.0076693310e-03
                                Analytic=6.9982100919e-03
                                iterations=57000
Energy [1]:
                                Numeric=2.8015845268e-02
                                Analytic=2.7982248553e-02
                                iterations=41000
Energy [2]:
                                Numeric=6.2986106101e-02
                                Analytic=6.2919369611e-02
                                iterations=26000
Energy [3]:
                                Numeric=1.1185553409e-01
                                Analytic=1.1175153189e-01
                                iterations=23000
Energy [4]:
                                Numeric=1.7454915460e-01
                                Analytic=1.7438928960e-01
                                iterations=17000
Energy [5]:
                                Numeric=2.5091722742e-01
                                Analytic=2.5070094967e-01
                                iterations=16000
Energy [6]:
                                Numeric=3.4077455143e-01
                                Analytic=3.4049319314e-01
                                iterations=13000
Energy [7]:
                                Numeric=4.4384102514e-01
                                Analytic=4.4347441113e-01
                                iterations=12000
Energy [8]:
                                Numeric=5.5960015499e-01
                                Analytic=5.5917793753e-01
                                iterations=11000
Energy [9]:
                                Numeric=6.8725503004e-01
                                Analytic=6.8677295883e-01
                                iterations=11000
Energy [10]:
                                Numeric=8.2502706438e-01
                                Analytic=8.2444718769e-01
                                iterations=10000
Energy [11]:
                                Numeric=9.6631814407e-01
                                Analytic=9.6542132955e-01
                                iterations=14000
Estamos em L = 69.39
Energy [0]:
                                Numeric=6.8744813930e-03
                                Analytic=6.8651461024e-03
                                iterations=58000
Energy [1]:
                                Numeric=2.7483012248e-02
                                Analytic=2.7450491746e-02
                                iterations=42000
Energy [2]:
                                Numeric=6.1788042394e-02
                                Analytic=6.1724852075e-02
                                iterations=27000
Energy [3]:
                                Numeric=1.0973869430e-01
                                Analytic=1.0963301777e-01
                                iterations=23000
Energy [4]:
                                Numeric=1.7124177057e-01
                                Analytic=1.7109007070e-01
                                iterations=18000
Energy [5]:
                                Numeric=2.4618481504e-01
                                Analytic=2.4597134236e-01
                                iterations=16000
Energy [6]:
                                Numeric=3.3438317896e-01
                                Analytic=3.3409459194e-01
                                iterations=13000
Energy [7]:
                                Numeric=4.3552507419e-01
                                Analytic=4.3518662193e-01
                                iterations=13000
Energy [8]:
                                Numeric=5.4922848540e-01
                                Analytic=5.4881439193e-01
                                iterations=11000
Energy [9]:
                                Numeric=6.7471324586e-01
                                Analytic=6.7421920309e-01
                                iterations=11000
Energy [10]:
                                Numeric=8.1036440382e-01
                                Analytic=8.0979789675e-01
                                iterations=10000
Energy [11]:
                                Numeric=9.5093787545e-01
                                Analytic=9.5017323257e-01
                                iterations=13000
Estamos em L = 70.10
Energy [0]:
                                Numeric=6.7450597318e-03
                                Analytic=6.7358410030e-03
                                iterations=59000
Energy [1]:
                                Numeric=2.6965272489e-02
                                Analytic=2.6933742572e-02
                                iterations=43000
Energy [2]:
                                Numeric=6.0628164737e-02
                                Analytic=6.0563992061e-02
                                iterations=27000
Energy [3]:
                                Numeric=1.0767429222e-01
                                Analytic=1.0757404704e-01
                                iterations=24000
Energy [4]:
                                Numeric=1.6803319826e-01
                                Analytic=1.6788324162e-01
                                iterations=18000
Energy [5]:
                                Numeric=2.4157800522e-01
                                Analytic=2.4137347839e-01
                                iterations=17000
Energy [6]:
                                Numeric=3.2813949062e-01
                                Analytic=2.6999663018e-01
                                iterations=14000
Energy [7]:
                                Numeric=4.2747011294e-01
                                Analytic=3.2787281897e-01
                                iterations=13000
Energy [8]:
                                Numeric=5.3915243328e-01
                                Analytic=4.2712503906e-01
                                iterations=11000
Energy [9]:
                                Numeric=6.6247154426e-01
                                Analytic=5.3872767941e-01
                                iterations=11000
Energy [10]:
                                Numeric=7.9602893128e-01
                                Analytic=6.6198613640e-01
                                iterations=10000
Energy [11]:
                                Numeric=9.3552250211e-01
                                Analytic=7.9547464888e-01
                                iterations=12000
Energy [12]:
                                Numeric=1.0027419194e+00
                                Analytic=9.3481291053e-01
                                iterations=102000
Estamos em L = 70.81
Energy [0]:
                                Numeric=6.6192637446e-03
                                Analytic=6.6101545936e-03
                                iterations=60000
Energy [1]:
                                Numeric=2.6464029504e-02
                                Analytic=2.6431441896e-02
                                iterations=43000
Energy [2]:
                                Numeric=5.9496416666e-02
                                Analytic=5.9435538712e-02
                                iterations=28000
Energy [3]:
                                Numeric=1.0567207493e-01
                                Analytic=1.0557241546e-01
                                iterations=24000
Energy [4]:
                                Numeric=1.6491826363e-01
                                Analytic=1.6476540270e-01
                                iterations=18000
Energy [5]:
                                Numeric=2.3710385128e-01
                                Analytic=2.3690255503e-01
                                iterations=17000
Energy [6]:
                                Numeric=3.2208343163e-01
                                Analytic=3.2182152512e-01
                                iterations=14000
Energy [7]:
                                Numeric=4.1962173606e-01
                                Analytic=4.1928175820e-01
                                iterations=13000
Energy [8]:
                                Numeric=5.2932769624e-01
                                Analytic=5.2890866549e-01
                                iterations=11000
Energy [9]:
                                Numeric=6.5054158511e-01
                                Analytic=6.5006481304e-01
                                iterations=11000
Energy [10]:
                                Numeric=7.8201846285e-01
                                Analytic=7.8147610287e-01
                                iterations=10000
Energy [11]:
                                Numeric=9.2019691463e-01
                                Analytic=9.1950425697e-01
                                iterations=11000
Estamos em L = 71.52
Energy [0]:
                                Numeric=6.4975128646e-03
                                Analytic=6.4879530696e-03
                                iterations=60000
Energy [1]:
                                Numeric=2.5974641338e-02
                                Analytic=2.5943056363e-02
                                iterations=44000
Energy [2]:
                                Numeric=5.8399578439e-02
                                Analytic=5.8338298639e-02
                                iterations=28000
Energy [3]:
                                Numeric=1.0372755405e-01
                                Analytic=1.0362601953e-01
                                iterations=24000
Energy [4]:
                                Numeric=1.6187478067e-01
                                Analytic=1.6173330816e-01
                                iterations=19000
Energy [5]:
                                Numeric=2.3275222367e-01
                                Analytic=2.3255398347e-01
                                iterations=17000
Energy [6]:
                                Numeric=3.1620228028e-01
                                Analytic=3.1593463603e-01
                                iterations=14000
Energy [7]:
                                Numeric=4.1198432540e-01
                                Analytic=4.1164919669e-01
                                iterations=13000
Energy [8]:
                                Numeric=5.1976230299e-01
                                Analytic=5.1934852724e-01
                                iterations=11000
Energy [9]:
                                Numeric=6.3891458950e-01
                                Analytic=6.3844635480e-01
                                iterations=11000
Energy [10]:
                                Numeric=7.6832992321e-01
                                Analytic=7.6779915199e-01
                                iterations=10000
Energy [11]:
                                Numeric=9.0497849305e-01
                                Analytic=9.0433900065e-01
                                iterations=11000
Estamos em L = 72.22
Energy [0]:
                                Numeric=6.3785626091e-03
                                Analytic=6.3691088436e-03
                                iterations=61000
Energy [1]:
                                Numeric=2.5498735636e-02
                                Analytic=2.5468076984e-02
                                iterations=45000
Energy [2]:
                                Numeric=5.7329340419e-02
                                Analytic=5.7271132791e-02
                                iterations=29000
Energy [3]:
                                Numeric=1.0182900450e-01
                                Analytic=1.0173285095e-01
                                iterations=25000
Energy [4]:
                                Numeric=1.5892765765e-01
                                Analytic=1.5878385783e-01
                                iterations=19000
Energy [5]:
                                Numeric=2.2852626067e-01
                                Analytic=2.2832337773e-01
                                iterations=17000
Energy [6]:
                                Numeric=3.1047082505e-01
                                Analytic=3.1020633860e-01
                                iterations=14000
Energy [7]:
                                Numeric=4.0455069941e-01
                                Analytic=4.0422008005e-01
                                iterations=13000
Energy [8]:
                                Numeric=5.1042038296e-01
                                Analytic=5.1003874852e-01
                                iterations=12000
Energy [9]:
                                Numeric=6.2758178779e-01
                                Analytic=6.2712199675e-01
                                iterations=11000
Energy [10]:
                                Numeric=7.5498539972e-01
                                Analytic=7.5443941701e-01
                                iterations=10000
Energy [11]:
                                Numeric=8.8997617973e-01
                                Analytic=8.8937227410e-01
                                iterations=11000
Estamos em L = 72.93
Energy [0]:
                                Numeric=6.2628546832e-03
                                Analytic=6.2535000222e-03
                                iterations=62000
Energy [1]:
                                Numeric=2.5037784364e-02
                                Analytic=2.5006017823e-02
                                iterations=45000
Energy [2]:
                                Numeric=5.6292075218e-02
                                Analytic=5.6232953515e-02
                                iterations=29000
Energy [3]:
                                Numeric=9.9986786199e-02
                                Analytic=9.9890991570e-02
                                iterations=25000
Energy [4]:
                                Numeric=1.5605670778e-01
                                Analytic=1.5591408947e-01
                                iterations=19000
Energy [5]:
                                Numeric=2.2439422036e-01
                                Analytic=2.2420654417e-01
                                iterations=18000
Energy [6]:
                                Numeric=3.0489259533e-01
                                Analytic=3.0463106825e-01
                                iterations=14000
Energy [7]:
                                Numeric=3.9730504777e-01
                                Analytic=3.9698742885e-01
                                iterations=14000
Energy [8]:
                                Numeric=5.0134487408e-01
                                Analytic=5.0097111393e-01
                                iterations=12000
Energy [9]:
                                Numeric=6.1655642356e-01
                                Analytic=6.1608312155e-01
                                iterations=11000
Energy [10]:
                                Numeric=7.4192937529e-01
                                Analytic=7.4139160400e-01
                                iterations=10000
Energy [11]:
                                Numeric=8.7521644991e-01
                                Analytic=8.7463843335e-01
                                iterations=11000
Energy [12]:
                                Numeric=1.0032689427e+00
                                Analytic=9.9986266841e-01
                                iterations=28000
Estamos em L = 73.64
Energy [0]:
                                Numeric=6.1502723960e-03
                                Analytic=6.1410102444e-03
                                iterations=63000
Energy [1]:
                                Numeric=2.4587241385e-02
                                Analytic=2.4556414736e-02
                                iterations=46000
Energy [2]:
                                Numeric=5.5282993440e-02
                                Analytic=5.5222721811e-02
                                iterations=29000
Energy [3]:
                                Numeric=9.8189635514e-02
                                Analytic=9.8098608584e-02
                                iterations=26000
Energy [4]:
                                Numeric=1.5326281343e-01
                                Analytic=1.5312117156e-01
                                iterations=19000
Energy [5]:
                                Numeric=2.2039069100e-01
                                Analytic=2.2019947167e-01
                                iterations=18000
Energy [6]:
                                Numeric=2.9944367741e-01
                                Analytic=2.9920349676e-01
                                iterations=15000
Energy [7]:
                                Numeric=3.9025715055e-01
                                Analytic=3.8994454604e-01
                                iterations=14000
Energy [8]:
                                Numeric=4.9252203910e-01
                                Analytic=4.9213770166e-01
                                iterations=12000
Energy [9]:
                                Numeric=6.0578909845e-01
                                Analytic=6.0532128489e-01
                                iterations=11000
Energy [10]:
                                Numeric=7.2917989537e-01
                                Analytic=7.2864976825e-01
                                iterations=10000
Energy [11]:
                                Numeric=8.6071610290e-01
                                Analytic=8.6015909890e-01
                                iterations=11000
Energy [12]:
                                Numeric=9.9147236927e-01
                                Analytic=9.9010191133e-01
                                iterations=19000
Estamos em L = 74.34
Energy [0]:
                                Numeric=6.0407042690e-03
                                Analytic=6.0315283633e-03
                                iterations=64000
Energy [1]:
                                Numeric=2.4148779746e-02
                                Analytic=2.4118824235e-02
                                iterations=47000
Energy [2]:
                                Numeric=5.4296567938e-02
                                Analytic=5.4239444740e-02
                                iterations=30000
Energy [3]:
                                Numeric=9.6446467737e-02
                                Analytic=9.6353950034e-02
                                iterations=26000
Energy [4]:
                                Numeric=1.5053688888e-01
                                Analytic=1.5040239647e-01
                                iterations=20000
Energy [5]:
                                Numeric=2.1648735144e-01
                                Analytic=2.1629832234e-01
                                iterations=18000
Energy [6]:
                                Numeric=2.9416481628e-01
                                Analytic=2.9391852080e-01
                                iterations=15000
Energy [7]:
                                Numeric=3.8339271235e-01
                                Analytic=3.8308500473e-01
                                iterations=14000
Energy [8]:
                                Numeric=4.8390978249e-01
                                Analytic=4.8353087576e-01
                                iterations=12000
Energy [9]:
                                Numeric=5.9526043231e-01
                                Analytic=5.9482823390e-01
                                iterations=12000
Energy [10]:
                                Numeric=7.1669771329e-01
                                Analytic=7.1620751149e-01
                                iterations=11000
Energy [11]:
                                Numeric=8.4651653972e-01
                                Analytic=8.4594764557e-01
                                iterations=11000
Energy [12]:
                                Numeric=9.7798345432e-01
                                Analytic=9.7695082777e-01
                                iterations=15000
Estamos em L = 75.05
Energy [0]:
                                Numeric=5.9340437582e-03
                                Analytic=5.9249480708e-03
                                iterations=65000
Energy [1]:
                                Numeric=2.3721967793e-02
                                Analytic=2.3692822380e-02
                                iterations=48000
Energy [2]:
                                Numeric=5.3340406589e-02
                                Analytic=5.3282173008e-02
                                iterations=30000
Energy [3]:
                                Numeric=9.4749661763e-02
                                Analytic=9.4655340585e-02
                                iterations=26000
Energy [4]:
                                Numeric=1.4788835043e-01
                                Analytic=1.4775517407e-01
                                iterations=20000
Energy [5]:
                                Numeric=2.1268645853e-01
                                Analytic=2.1249942274e-01
                                iterations=18000
Energy [6]:
                                Numeric=2.8901427040e-01
                                Analytic=2.8877125103e-01
                                iterations=15000
Energy [7]:
                                Numeric=3.7670561841e-01
                                Analytic=3.7640263638e-01
                                iterations=14000
Energy [8]:
                                Numeric=4.7551685217e-01
                                Analytic=4.7514327781e-01
                                iterations=12000
Energy [9]:
                                Numeric=5.8501995242e-01
                                Analytic=5.8459592017e-01
                                iterations=12000
Energy [10]:
                                Numeric=7.0453838808e-01
                                Analytic=7.0405813083e-01
                                iterations=11000
Energy [11]:
                                Numeric=8.3256752636e-01
                                Analytic=8.3201189078e-01
                                iterations=11000
Energy [12]:
                                Numeric=9.6383609179e-01
                                Analytic=9.6296595183e-01
                                iterations=13000
Estamos em L = 75.76
Energy [0]:
                                Numeric=5.8296815137e-03
                                Analytic=5.8211677395e-03
                                iterations=67000
Energy [1]:
                                Numeric=2.3308236648e-02
                                Analytic=2.3278003751e-02
                                iterations=48000
Energy [2]:
                                Numeric=5.2405757668e-02
                                Analytic=5.2349998680e-02
                                iterations=31000
Energy [3]:
                                Numeric=9.3088773801e-02
                                Analytic=9.3001177553e-02
                                iterations=27000
Energy [4]:
                                Numeric=1.4531328628e-01
                                Analytic=1.4517702572e-01
                                iterations=20000
Energy [5]:
                                Numeric=2.0897804291e-01
                                Analytic=2.0879925568e-01
                                iterations=19000
Energy [6]:
                                Numeric=2.8399703251e-01
                                Analytic=2.8375700181e-01
                                iterations=15000
Energy [7]:
                                Numeric=3.7020323951e-01
                                Analytic=3.6989151951e-01
                                iterations=14000
Energy [8]:
                                Numeric=4.6733628947e-01
                                Analytic=4.6696781834e-01
                                iterations=12000
Energy [9]:
                                Numeric=5.7503261930e-01
                                Analytic=5.7461650844e-01
                                iterations=12000
Energy [10]:
                                Numeric=6.9266501314e-01
                                Analytic=6.9219473212e-01
                                iterations=11000
Energy [11]:
                                Numeric=8.1889926163e-01
                                Analytic=8.1835579327e-01
                                iterations=11000
Energy [12]:
                                Numeric=9.4946558755e-01
                                Analytic=9.4869977365e-01
                                iterations=12000
Estamos em L = 76.46
Energy [0]:
                                Numeric=5.7285405632e-03
                                Analytic=5.7200901620e-03
                                iterations=68000
Energy [1]:
                                Numeric=2.2903390023e-02
                                Analytic=2.2873980497e-02
                                iterations=49000
Energy [2]:
                                Numeric=5.1498884381e-02
                                Analytic=5.1442053047e-02
                                iterations=31000
Energy [3]:
                                Numeric=9.1479231123e-02
                                Analytic=9.1389927172e-02
                                iterations=27000
Energy [4]:
                                Numeric=1.4279141355e-01
                                Analytic=1.4266557861e-01
                                iterations=21000
Energy [5]:
                                Numeric=2.0537085674e-01
                                Analytic=2.0519445238e-01
                                iterations=19000
Energy [6]:
                                Numeric=2.7910171835e-01
                                Analytic=2.7887128140e-01
                                iterations=16000
Energy [7]:
                                Numeric=3.6383411156e-01
                                Analytic=3.6354596881e-01
                                iterations=15000
Energy [8]:
                                Numeric=4.5935242954e-01
                                Analytic=4.5899766803e-01
                                iterations=13000
Energy [9]:
                                Numeric=5.6531109009e-01
                                Analytic=5.6488238195e-01
                                iterations=12000
Energy [10]:
                                Numeric=6.8109575442e-01
                                Analytic=6.8061031678e-01
                                iterations=11000
Energy [11]:
                                Numeric=8.0551278812e-01
                                Analytic=8.0498057321e-01
                                iterations=11000
Energy [12]:
                                Numeric=9.3504540574e-01
                                Analytic=9.3438206241e-01
                                iterations=12000
Estamos em L = 77.17
Energy [0]:
                                Numeric=5.6300141810e-03
                                Analytic=5.6216223486e-03
                                iterations=69000
Energy [1]:
                                Numeric=2.2509023471e-02
                                Analytic=2.2480381430e-02
                                iterations=50000
Energy [2]:
                                Numeric=5.0611528211e-02
                                Analytic=5.0557504611e-02
                                iterations=32000
Energy [3]:
                                Numeric=8.9905011284e-02
                                Analytic=8.9820121083e-02
                                iterations=28000
Energy [4]:
                                Numeric=1.4034699186e-01
                                Analytic=1.4021856039e-01
                                iterations=21000
Energy [5]:
                                Numeric=2.0186230159e-01
                                Analytic=2.0168178516e-01
                                iterations=19000
Energy [6]:
                                Numeric=2.7433685877e-01
                                Analytic=2.7410978271e-01
                                iterations=16000
Energy [7]:
                                Numeric=3.5764386519e-01
                                Analytic=3.5736052474e-01
                                iterations=15000
Energy [8]:
                                Numeric=4.5157548918e-01
                                Analytic=4.5122624886e-01
                                iterations=13000
Energy [9]:
                                Numeric=5.5580933453e-01
                                Analytic=5.5538614511e-01
                                iterations=12000
Energy [10]:
                                Numeric=6.6977599047e-01
                                Analytic=6.6929784859e-01
                                iterations=11000
Energy [11]:
                                Numeric=7.9240662346e-01
                                Analytic=7.9188547664e-01
                                iterations=11000
Energy [12]:
                                Numeric=9.2078679737e-01
                                Analytic=9.2013179137e-01
                                iterations=11000
Estamos em L = 77.88
Energy [0]:
                                Numeric=5.5340131470e-03
                                Analytic=5.5256751446e-03
                                iterations=70000
Energy [1]:
                                Numeric=2.2124775860e-02
                                Analytic=2.2096851169e-02
                                iterations=51000
Energy [2]:
                                Numeric=4.9750631569e-02
                                Analytic=4.9695557190e-02
                                iterations=32000
Energy [3]:
                                Numeric=8.8376759811e-02
                                Analytic=8.8290353024e-02
                                iterations=28000
Energy [4]:
                                Numeric=1.3796139774e-01
                                Analytic=1.3783379417e-01
                                iterations=21000
Energy [5]:
                                Numeric=1.9843741359e-01
                                Analytic=1.9825816048e-01
                                iterations=19000
Energy [6]:
                                Numeric=2.6969216678e-01
                                Analytic=2.6946837453e-01
                                iterations=16000
Energy [7]:
                                Numeric=3.5160852741e-01
                                Analytic=3.5132994358e-01
                                iterations=15000
Energy [8]:
                                Numeric=4.4399110966e-01
                                Analytic=4.4364722527e-01
                                iterations=13000
Energy [9]:
                                Numeric=5.4653854378e-01
                                Analytic=5.4612062415e-01
                                iterations=12000
Energy [10]:
                                Numeric=6.5872131427e-01
                                Analytic=6.5825030506e-01
                                iterations=11000
Energy [11]:
                                Numeric=7.7957862551e-01
                                Analytic=7.7906831209e-01
                                iterations=11000
Energy [12]:
                                Numeric=9.0662609557e-01
                                Analytic=9.0601779752e-01
                                iterations=11000
Estamos em L = 78.59
Energy [0]:
                                Numeric=5.4404520194e-03
                                Analytic=5.4321633058e-03
                                iterations=71000
Energy [1]:
                                Numeric=2.1752061138e-02
                                Analytic=2.1723049337e-02
                                iterations=51000
Energy [2]:
                                Numeric=4.8907856278e-02
                                Analytic=4.8855448139e-02
                                iterations=33000
Energy [3]:
                                Numeric=8.6885758187e-02
                                Analytic=8.6799275718e-02
                                iterations=28000
Energy [4]:
                                Numeric=1.3563074847e-01
                                Analytic=1.3550919379e-01
                                iterations=22000
Energy [5]:
                                Numeric=1.9508538888e-01
                                Analytic=1.9492061242e-01
                                iterations=20000
Energy [6]:
                                Numeric=2.6517289133e-01
                                Analytic=2.6494309318e-01
                                iterations=16000
Energy [7]:
                                Numeric=3.4573636370e-01
                                Analytic=3.4544918786e-01
                                iterations=15000
Energy [8]:
                                Numeric=4.3659336722e-01
                                Analytic=4.3625449538e-01
                                iterations=13000
Energy [9]:
                                Numeric=5.3746451599e-01
                                Analytic=5.3707886602e-01
                                iterations=13000
Energy [10]:
                                Numeric=6.4792475850e-01
                                Analytic=6.4746071705e-01
                                iterations=11000
Energy [11]:
                                Numeric=7.6705248490e-01
                                Analytic=7.6652583595e-01
                                iterations=11000
Energy [12]:
                                Numeric=8.9265853896e-01
                                Analytic=8.9208243070e-01
                                iterations=11000
Estamos em L = 79.29
Energy [0]:
                                Numeric=5.3492489436e-03
                                Analytic=5.3410051559e-03
                                iterations=72000
Energy [1]:
                                Numeric=2.1386931453e-02
                                Analytic=2.1358649816e-02
                                iterations=52000
Energy [2]:
                                Numeric=4.8089890176e-02
                                Analytic=4.8036446670e-02
                                iterations=33000
Energy [3]:
                                Numeric=8.5427628968e-02
                                Analytic=8.5345597938e-02
                                iterations=29000
Energy [4]:
                                Numeric=1.3336322337e-01
                                Analytic=1.3324275933e-01
                                iterations=22000
Energy [5]:
                                Numeric=1.9183524496e-01
                                Analytic=1.9166629646e-01
                                iterations=20000
Energy [6]:
                                Numeric=2.6075797944e-01
                                Analytic=2.6053013467e-01
                                iterations=16000
Energy [7]:
                                Numeric=3.3999773210e-01
                                Analytic=3.3971341731e-01
                                iterations=15000
Energy [8]:
                                Numeric=4.2937605996e-01
                                Analytic=4.2904218235e-01
                                iterations=13000
Energy [9]:
                                Numeric=5.2863206523e-01
                                Analytic=5.2825413616e-01
                                iterations=13000
Energy [10]:
                                Numeric=6.3737943553e-01
                                Analytic=6.3692219911e-01
                                iterations=11000
Energy [11]:
                                Numeric=7.5477297785e-01
                                Analytic=7.5425403472e-01
                                iterations=11000
Energy [12]:
                                Numeric=8.7890454306e-01
                                Analytic=8.7835257568e-01
                                iterations=11000
Energy [13]:
                                Numeric=9.9979896431e-01
                                Analytic=9.9766321067e-01
                                iterations=24000
Estamos em L = 80.00
In [23]:
L,e=tuple(zip(*eigenvalues))
l_e = pd.DataFrame({'L':L, 'e':e})
l_e.to_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv')
In [5]:
numeric = pd.read_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv', index_col=0)
In [6]:
ax = numeric[(numeric['L'] <= 80) & (numeric['e'] <= 1.0)].plot.scatter(x='L',y='e')
ax.set_xlabel(r'a (\AA)')
ax.set_ylabel(r'E (eV)')
ax.set_ylim([-0.05, 1.05])
ax.set_xlim([8, 82])
plt.savefig('figuras/poco_quantico_numerico.png', bbox_inches='tight')
In [7]:
analitic = pd.read_csv('saidas/poco_quantico_autovalores_por_comprimento.csv', index_col=0)
ax = analitic[analitic['L'] <= 80].plot.scatter(x='L',y='e')
ax.set_xlabel(r'a (\AA)')
ax.set_ylabel(r'E (eV)')
ax.set_ylim([-0.05, 1.05])
ax.set_xlim([8, 82])
plt.savefig('figuras/poco_quantico_analitico.png', bbox_inches='tight')
In [8]:
numcut = pd.read_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv', index_col=0)
numcut = numcut[(numcut['L'] <= 80) & (numcut['e'] <= 1.0)]
anacut = pd.read_csv('saidas/poco_quantico_autovalores_por_comprimento.csv', index_col=0)
anacut = anacut[(anacut['L'] <= 80) & (anacut['e'] <= 1.0)]
In [9]:
numcut['type'] = ['num']*numcut.shape[0]
anacut['type'] = ['ana']*anacut.shape[0]
In [10]:
fig, ax = plt.subplots(1,1)
ax.scatter(anacut['L'], anacut['e'], marker='o',c='k', s=40)
ax.scatter(numcut['L'], numcut['e'], marker='o',c='w', s=20)
ax.set_xlabel(r'a (\AA)')
ax.set_ylabel(r'E (eV)')
ax.set_ylim([-0.05, 1.05])
ax.set_xlim([8, 82])
plt.savefig('figuras/poco_quantico_numerico_e_analitico.png', bbox_inches='tight')

Poço de 50 A

In [11]:
poco_50 = PocoQuantico(well_length=50.0, well_height=1.0, N=2048, dt=1e-18)
In [12]:
fix, ax = plt.subplots(1,1, figsize=(18,6))
ax.plot(poco_50.z_ang, poco_50.v_ev)
ax.set_xlabel(r'z (A)')
ax.set_ylabel(r'V (eV)')
plt.show()
In [13]:
poco_50.evolucao_imaginaria(precision=1e-4)
Energy [0]:
                                Numeric=1.2951868994e-02
                                Analytic=1.2937397611e-02
                                iterations=33000
Energy [1]:
                                Numeric=5.1752222957e-02
                                Analytic=5.1699753783e-02
                                iterations=24000
Energy [2]:
                                Numeric=1.1623530904e-01
                                Analytic=1.1612905356e-01
                                iterations=16000
Energy [3]:
                                Numeric=2.0610841151e-01
                                Analytic=2.0592968138e-01
                                iterations=14000
Energy [4]:
                                Numeric=3.2087362193e-01
                                Analytic=3.2060263426e-01
                                iterations=11000
Energy [5]:
                                Numeric=4.5967820296e-01
                                Analytic=4.5929848646e-01
                                iterations=10000
Energy [6]:
                                Numeric=6.2092866745e-01
                                Analytic=6.2043969392e-01
                                iterations=9000
Energy [7]:
                                Numeric=8.0102193683e-01
                                Analytic=8.0041283309e-01
                                iterations=9000
Energy [8]:
                                Numeric=9.8405183319e-01
                                Analytic=9.8285878862e-01
                                iterations=18000
Out[13]:
<core.poco_quadrado_finito.PocoQuantico at 0x7f0dbe5b1940>
In [14]:
fix, ax = plt.subplots(1,1)
vsize = np.ptp(poco_50.v_ev)
ax.plot(poco_50.z_ang, poco_50.v_ev, label=r'$V(z)$')

for i, es in enumerate(poco_50.eigenstates):
    psi2 = np.abs(es)**2
    psi2 = 0.02 * psi2 * vsize / np.max(psi2) + poco_50.eigenvalues[i]
    ax.plot(poco_50.z_ang, psi2, label=r'$|\psi_{%d} (z)|^2$' % i)
    
ax.set_xlabel(r'$z$ (\AA)')
ax.set_ylabel(r'$E$ (eV)')
plt.legend(bbox_to_anchor=(0,1.02,1,0.2), loc="lower left",
                mode="expand", borderaxespad=0, ncol=5)
plt.savefig('figuras/poco_quantico_numerico_autoestados.png', bbox_inches='tight')